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整数序列中的多种数学数表解析
在数学的奇妙世界里，存在着众多有趣且重要的整数序列和数表。这些数表涵盖了各种类型的数学数，如斯特林数、欧拉数、贝尔数等，它们在组合数学、数论等领域有着广泛的应用。
斯特林数相关数表
斯特林数分为第一类和第二类，它们在组合数学中描述了不同的组合方式。
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第一类斯特林数（A132393）
：用于计算将 (n) 个元素排列成 (k) 个循环排列的方式数。例如，当 (n = 3)，(k = 2) 时，第一类斯特林数为 (3)，这表示将 (3) 个元素排列成 (2) 个循环排列有 (3) 种不同的方式。其数表如下：
| (n \diagdown k) | (k = 1) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) | (k = 9) |
| — | — | — | — | — | — | — | — | — | — |
| (n = 1) | (1) | | | | | | | | |
| (n = 2) | (1) | (1) | | | | | | | |
| (n = 3) | (2) | (3) | (1) | | | | | | |
| (n = 4) | (6) | (11) | (6) | (1) | | | | | |
| (n = 5) | (24) | (50) | (35) | (10) | (1) | | | | |
| (n = 6) | (120) | (274) | (225) | (85) | (15) | (1) | | | |
| (n = 7) | (720) | (1764) | (1624) | (735) | (175) | (21) | (1) | | |
| (n = 8) | (5040) | (13068) | (13132) | (6769) | (1960) | (322) | (28) | (1) | |
| (n = 9) | (40320) | (109584) | (118124) | (67284) | (22449) | (4536) | (546) | (36) | (1) |
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第二类斯特林数（A008277）
：用于计算将 (n) 个元素划分成 (k) 个非空子集的方式数。例如，当 (n = 3)，(k = 2) 时，第二类斯特林数为 (3)，意味着将 (3) 个元素划分成 (2) 个非空子集有 (3) 种不同的划分方法。其数表如下：
| (n \diagdown k) | (k = 1) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) | (k = 9) |
| — | — | — | — | — | — | — | — | — | — |
| (n = 1) | (1) | | | | | | | | |
| (n = 2) | (1) | (1) | | | | | | | |
| (n = 3) | (1) | (3) | (1) | | | | | | |
| (n = 4) | (1) | (7) | (6) | (1) | | | | | |
| (n = 5) | (1) | (15) | (25) | (10) | (1) | | | | |
| (n = 6) | (1) | (31) | (90) | (65) | (15) | (1) | | | |
| (n = 7) | (1) | (63) | (301) | (350) | (140) | (21) | (1) | | |
| (n = 8) | (1) | (127) | (966) | (1701) | (1050) | (266) | (28) | (1) | |
| (n = 9) | (1) | (255) | (3025) | (7770) | (6951) | (2646) | (462) | (36) | (1) |
除了普通的斯特林数，还有 (2 -) 斯特林数和 (3 -) 斯特林数。
-
2 - 第一类斯特林数（A143491）
：在某些特定的组合问题中有独特的应用。其数表展示了不同 (n) 和 (k) 值下的对应数值。
| (n \diagdown k) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) | (k = 9) |
| — | — | — | — | — | — | — | — | — |
| (n = 2) | (1) | | | | | | | |
| (n = 3) | (2) | (1) | | | | | | |
| (n = 4) | (6) | (5) | (1) | | | | | |
| (n = 5) | (24) | (26) | (9) | (1) | | | | |
| (n = 6) | (120) | (154) | (71) | (14) | (1) | | | |
| (n = 7) | (720) | (1044) | (580) | (155) | (20) | (1) | | |
| (n = 8) | (5040) | (8028) | (5104) | (1665) | (295) | (27) | (1) | |
| (n = 9) | (40320) | (69264) | (48860) | (18424) | (4025) | (511) | (35) | (1) |
| (n = 10) | (362880) | (663696) | (509004) | (214676) | (54649) | (8624) | (826) | (44) |
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2 - 第二类斯特林数（A143494） ：同样在组合数学的特定场景中发挥作用。
| (n \diagdown k) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) | (k = 9) | (k = 10) |
| — | — | — | — | — | — | — | — | — | — |
| (n = 2) | (1) | | | | | | | | |
| (n = 3) | (2) | (1) | | | | | | | |
| (n = 4) | (4) | (5) | (1) | | | | | | |
| (n = 5) | (8) | (19) | (9) | (1) | | | | | |
| (n = 6) | (16) | (65) | (55) | (14) | (1) | | | | |
| (n = 7) | (32) | (211) | (285) | (125) | (20) | (1) | | | |
| (n = 8) | (64) | (665) | (1351) | (910) | (245) | (27) | (1) | | |
| (n = 9) | (128) | (2059) | (6069) | (5901) | (2380) | (434) | (35) | (1) | |
| (n = 10) | (256) | (6305) | (26335) | (35574) | (20181) | (5418) | (714) | (44) | (1) | -
3 - 第一类斯特林数（A143492） ：在更复杂的组合问题建模中具有重要意义。
| (n \diagdown k) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) | (k = 9) | (k = 10) |
| — | — | — | — | — | — | — | — | — |
| (n = 3) | (1) | | | | | | | |
| (n = 4) | (3) | (1) | | | | | | |
| (n = 5) | (12) | (7) | (1) | | | | | |
| (n = 6) | (60) | (47) | (12) | (1) | | | | |
| (n = 7) | (360) | (342) | (119) | (18) | (1) | | | |
| (n = 8) | (2520) | (2754) | (1175) | (245) | (25) | (1) | | |
| (n = 9) | (20160) | (24552) | (12154) | (3135) | (445) | (33) | (1) | |
| (n = 10) | (181440) | (241128) | (133938) | (40369) | (7140) | (742) | (42) | (1) |
| (n = 11) | (1814400) | (2592720) | (1580508) | (537628) | (111769) | (14560) | (1162) | (52) | -
3 - 第二类斯特林数（A143495） ：在相关的组合划分问题中有其独特的应用价值。
| (n \diagdown k) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) | (k = 9) | (k = 10) |
| — | — | — | — | — | — | — | — | — |
| (n = 3) | (1) | | | | | | | |
| (n = 4) | (3) | (1) | | | | | | |
| (n = 5) | (9) | (7) | (1) | | | | | |
| (n = 6) | (27) | (37) | (12) | (1) | | | | |
| (n = 7) | (81) | (175) | (97) | (18) | (1) | | | |
| (n = 8) | (243) | (781) | (660) | (205) | (25) | (1) | | |
| (n = 9) | (729) | (3367) | (4081) | (1890) | (380) | (33) | (1) | |
| (n = 10) | (2187) | (14197) | (23772) | (15421) | (4550) | (644) | (42) | (1) |
| (n = 11) | (6561) | (58975) | (133057) | (116298) | (47271) | (9702) | (1022) | (52) |
此外，还有 (2 -) 受限斯特林数、(3 -) 受限斯特林数、(2 -) 关联斯特林数和 (3 -) 关联斯特林数等。这些数在不同的组合场景中有着各自独特的意义和应用。
欧拉数相关数表
欧拉数在数学分析、组合数学等领域有着重要的地位。
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欧拉数（A008292）
：描述了排列中上升和下降模式的数量。其数表如下：
| (n \diagdown k) | (k = 1) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) | (k = 9) |
| — | — | — | — | — | — | — | — | — | — |
| (n = 1) | (0) | | | | | | | | |
| (n = 2) | (1) | (0) | | | | | | | |
| (n = 3) | (4) | (1) | (0) | | | | | | |
| (n = 4) | (11) | (11) | (1) | (0) | | | | | |
| (n = 5) | (26) | (66) | (26) | (1) | (0) | | | | |
| (n = 6) | (57) | (302) | (302) | (57) | (1) | (0) | | | |
| (n = 7) | (120) | (1191) | (2416) | (1191) | (120) | (1) | (0) | | |
| (n = 8) | (247) | (4293) | (15619) | (15619) | (4293) | (247) | (1) | (0) | |
| (n = 9) | (502) | (14608) | (88234) | (156190) | (88234) | (14608) | (502) | (1) | (0) |
-
2 - 欧拉数 ：在一些特定的数学模型中有其独特的应用。
| (n \diagdown k) | (k = 0) | (k = 1) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) |
| — | — | — | — | — | — | — | — |
| (n = 0) | (2) | | | | | | |
| (n = 1) | (2) | (4) | | | | | |
| (n = 2) | (2) | (14) | (8) | | | | |
| (n = 3) | (2) | (36) | (66) | (16) | | | |
| (n = 4) | (2) | (82) | (342) | (262) | (32) | | |
| (n = 5) | (2) | (176) | (1436) | (2416) | (946) | (64) | |
| (n = 6) | (2) | (366) | (5364) | (16844) | (14394) | (3222) | (128) |
| (n = 7) | (2) | (748) | (18654) | (99560) | (156190) | (76908) | (10562) |
| (n = 8) | (2) | (1514) | (61946) | (528818) | (1378310) | (1242398) | (33734) |
| (n = 9) | (2) | (3048) | (199464) | (2610840) | (10593276) | (15724248) | (8882952) | -
3 - 欧拉数 ：同样在特定的数学分析和组合问题中有其作用。
| (n \diagdown k) | (k = 0) | (k = 1) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) |
| — | — | — | — | — | — | — | — | — |
| (n = 0) | (6) | | | | | | | |
| (n = 1) | (6) | (18) | | | | | | |
| (n = 2) | (6) | (60) | (54) | | | | | |
| (n = 3) | (6) | (150) | (402) | (162) | | | | |
| (n = 4) | (6) | (336) | (1956) | (2256) | (486) | | | |
| (n = 5) | (6) | (714) | (7884) | (18804) | (11454) | (1458) | | |
| (n = 6) | (6) | (1476) | (28650) | (122520) | (151290) | (54564) | (4374) | |
| (n = 7) | (6) | (3006) | (97758) | (690630) | (1491570) | (1083834) | (248874) | (13122) |
贝尔数相关数表
贝尔数（A000110）用于计算集合的划分方式总数。例如，(B_3 = 5) 表示将一个包含 (3) 个元素的集合进行划分，一共有 (5) 种不同的划分方式。其数表如下：
| (n) | (B_n) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (2) |
| (3) | (5) |
| (4) | (15) |
| (5) | (52) |
| (6) | (203) |
| (7) | (877) |
| (8) | (4140) |
| (9) | (21147) |
| (10) | (115975) |
| (11) | (678570) |
| (12) | (4213597) |
| (13) | (27644437) |
| (14) | (190899322) |
| (15) | (1382958545) |
| (16) | (10480142147) |
| (17) | (82864869804) |
| (18) | (682076806159) |
| (19) | (5832742205057) |
| (20) | (51724158235372) |
| (21) | (474869816156751) |
| (22) | (4506715738447323) |
| (23) | (44152005855084346) |
| (24) | (445958869294805289) |
| (25) | (4638590332229999353) |
| (26) | (49631246523618756274) |
| (27) | (545717047936059989389) |
| (28) | (6160539404599934652455) |
| (29) | (71339801938860275191172) |
| (30) | (846749014511809332450147) |
富比尼数相关数表
富比尼数（A000670）也称为有序贝尔数，与贝尔数有一定的关联。其数表如下：
| (n) | (F_n) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (3) |
| (3) | (13) |
| (4) | (75) |
| (5) | (541) |
| (6) | (4683) |
| (7) | (47293) |
| (8) | (545835) |
| (9) | (7087261) |
| (10) | (102247563) |
| (11) | (1622632573) |
| (12) | (28091567595) |
| (13) | (526858348381) |
| (14) | (10641342970443) |
| (15) | (230283190977853) |
| (16) | (5315654681981355) |
| (17) | (130370767029135901) |
| (18) | (3385534663256845323) |
| (19) | (92801587319328411133) |
| (20) | (2677687796244384203115) |
| (21) | (81124824998504073881821) |
| (22) | (2574844419803190384544203) |
| (23) | (85438451336745709294580413) |
| (24) | (2958279121074145472650648875) |
| (25) | (106697365438475775825583498141) |
| (26) | (4002225759844168492486127539083) |
| (27) | (155897763918621623249276226253693) |
| (28) | (6297562064950066033518373935334635) |
| (29) | (263478385263023690020893329044576861) |
伯努利数相关数表
伯努利数在数论、分析等领域有着广泛的应用。奇数索引的伯努利数除了 (B_1 = -\frac{1}{2}) 外都为 (0)。其数表如下：
| (n) | (B_n) |
| — | — |
| (0) | (1) |
| (1) | (-\frac{1}{2}) |
| (2) | (\frac{1}{6}) |
| (4) | (-\frac{1}{30}) |
| (6) | (\frac{1}{42}) |
| (8) | (-\frac{1}{30}) |
| (10) | (\frac{5}{66}) |
| (12) | (-\frac{691}{2730}) |
| (14) | (\frac{7}{6}) |
| (16) | (-\frac{3617}{510}) |
| (18) | (\frac{43867}{798}) |
| (20) | (-\frac{174611}{330}) |
| (22) | (\frac{854513}{138}) |
| (24) | (-\frac{236364091}{2730}) |
| (26) | (\frac{8553103}{6}) |
| (28) | (-\frac{23749461029}{870}) |
| (30) | (\frac{8615841276005}{14322}) |
| (32) | (-\frac{7709321041217}{510}) |
| (34) | (\frac{2577687858367}{6}) |
| (36) | (-\frac{26315271553053477373}{1919190}) |
柯西数相关数表
柯西数分为第一类和第二类，它们在数学分析等领域有重要作用。
-
第一类柯西数
：其数表如下：
| (n) | (c_n) |
| — | — |
| (0) | (1) |
| (1) | (\frac{1}{2}) |
| (2) | (-\frac{1}{6}) |
| (3) | (\frac{1}{4}) |
| (4) | (-\frac{19}{30}) |
| (5) | (\frac{9}{4}) |
| (6) | (-\frac{863}{84}) |
| (7) | (\frac{1375}{24}) |
| (8) | (-\frac{33953}{90}) |
| (9) | (\frac{57281}{20}) |
| (10) | (-\frac{3250433}{132}) |
| (11) | (\frac{1891755}{8}) |
| (12) | (-\frac{13695779093}{5460}) |
| (13) | (\frac{24466579093}{840}) |
| (14) | (-\frac{132282840127}{360}) |
| (15) | (\frac{240208245823}{48}) |
| (16) | (-\frac{111956703448001}{1530}) |
| (17) | (\frac{4573423873125}{4}) |
| (18) | (-\frac{30342376302478019}{1596}) |
| (19) | (\frac{56310194579604163}{168}) |
-
第二类柯西数
：其数表如下：
| (n) | (C_n) |
| — | — |
| (0) | (1) |
| (1) | (\frac{1}{2}) |
| (2) | (\frac{5}{6}) |
| (3) | (\frac{9}{4}) |
| (4) | (\frac{251}{30}) |
| (5) | (\frac{475}{12}) |
| (6) | (\frac{19087}{84}) |
| (7) | (\frac{36799}{24}) |
| (8) | (\frac{1070017}{90}) |
| (9) | (\frac{2082753}{20}) |
| (10) | (\frac{134211265}{132}) |
| (11) | (\frac{262747265}{24}) |
| (12) | (\frac{703604254357}{5460}) |
| (13) | (\frac{1382741929621}{840}) |
| (14) | (\frac{8164168737599}{360}) |
| (15) | (\frac{5362709743125}{16}) |
| (16) | (\frac{8092989203533249}{1530}) |
| (17) | (\frac{15980174332775873}{180}) |
| (18) | (\frac{12600467236042756559}{7980}) |
| (19) | (\frac{24919383499187492303}{840}) |
其他数表
除了上述数表外，还有 (2 -) 富比尼数、(3 -) 富比尼数、阶乘数、调和与超调和数、幂等数、对合数、受限贝尔数、受限阶乘数、关联贝尔数、关联阶乘数、拉赫数、超阶乘数等数表。这些数表在不同的数学领域和实际应用中都有着各自独特的作用。
数表间的关系与应用流程
这些数表之间存在着复杂而有趣的关系，并且在实际应用中有着一定的流程。例如，在组合数学中，斯特林数和贝尔数可以用于解决集合的划分和排列问题。以下是一个简单的流程图，展示了在解决组合问题时，如何根据问题的特点选择合适的数表：
graph TD;
A[组合问题] --> B{问题类型};
B -->|集合划分| C[贝尔数、斯特林数];
B -->|排列模式| D[欧拉数];
B -->|其他组合场景| E[其他相关数表];
C --> F[根据数表计算结果];
D --> F;
E --> F;
F --> G[得出解决方案];
这些丰富的数表为我们解决各种数学问题提供了强大的工具，它们的性质和相互关系值得我们深入研究和探索。通过对这些数表的了解和应用，我们可以更好地理解组合数学、数论等领域的奥秘，为解决实际问题提供有力的支持。
其他重要数表解析
富比尼数扩展数表
除了普通的富比尼数，还有 (2 -) 富比尼数和 (3 -) 富比尼数。
-
2 - 富比尼数（A232472）
：在一些特定的组合计数问题中有独特应用。其数表如下：
| (n) | (F_{n,2}) |
| — | — |
| (1) | (10) |
| (2) | (62) |
| (3) | (466) |
| (4) | (4142) |
| (5) | (42610) |
| (6) | (498542) |
| (7) | (6541426) |
| (8) | (95160302) |
| (9) | (1520385010) |
| (10) | (26468935022) |
| (11) | (498766780786) |
| (12) | (10114484622062) |
| (13) | (219641848007410) |
| (14) | (5085371491003502) |
| (15) | (125055112347154546) |
| (16) | (3255163896227709422) |
| (17) | (89416052656071565810) |
| (18) | (2584886208925055791982) |
| (19) | (78447137202259689678706) |
| (20) | (2493719594804686310662382) |
| (21) | (82863606916942518910036210) |
| (22) | (2872840669737399763356068462) |
| (23) | (103739086317401630352932849266) |
| (24) | (3895528394405692716660544040942) |
| (25) | (151895538158777454756790098714610) |
| (26) | (6141664301031444410269097709080942) |
| (27) | (257180823198073623987374955109242226) |
| (28) | (11140090408748856793721570867140325102) |
| (29) | (498604467780257507947098559879733217010) |
| (30) | (23035146049160627260753649613068937357422) |
-
3 - 富比尼数（A232473）
：在更复杂的组合场景中有其价值。
| (n) | (F_{n,3}) |
| — | — |
| (1) | (42) |
| (2) | (342) |
| (3) | (3210) |
| (4) | (34326) |
| (5) | (413322) |
| (6) | (5544342) |
| (7) | (82077450) |
| (8) | (1330064406) |
| (9) | (23428165002) |
| (10) | (445828910742) |
| (11) | (9116951060490) |
| (12) | (199412878763286) |
| (13) | (4646087794988682) |
| (14) | (114884369365147542) |
| (15) | (3005053671533400330) |
| (16) | (82905724863616146966) |
| (17) | (2406054103612912660362) |
| (18) | (73277364784409578094742) |
| (19) | (2336825320400166931304970) |
| (20) | (77876167727333146288711446) |
| (21) | (2707113455903514725535996042) |
| (22) | (97993404977926830826220712342) |
| (23) | (3688050221770889455954678342410) |
| (24) | (144104481369966069323469010632726) |
| (25) | (5837873224713889500755517511651722) |
| (26) | (244897494596010735166836759691080342) |
| (27) | (10625728762352709545746820956921840650) |
| (28) | (476324286962759794359655418145452566806) |
| (29) | (22037937113600112244859452493309470923402) |
| (30) | (1051344296019000117096802419429724152398742) |
阶乘数表
阶乘数（A000142）在排列组合问题中非常基础且重要。其数表如下：
| (n) | (n!) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (2) |
| (3) | (6) |
| (4) | (24) |
| (5) | (120) |
| (6) | (720) |
| (7) | (5040) |
| (8) | (40320) |
| (9) | (362880) |
| (10) | (3628800) |
| (11) | (39916800) |
| (12) | (479001600) |
| (13) | (6227020800) |
| (14) | (87178291200) |
| (15) | (1307674368000) |
| (16) | (20922789888000) |
| (17) | (355687428096000) |
| (18) | (6402373705728000) |
| (19) | (121645100408832000) |
| (20) | (2432902008176640000) |
| (21) | (51090942171709440000) |
| (22) | (11240007277776076800000) |
| (23) | (258520167388849766400000) |
| (24) | (6204484017332394393600000) |
| (25) | (1551121004333098598400000000) |
| (26) | (40329146112660563558400000000) |
| (27) | (1088886945041835216076800000000) |
| (28) | (30488834461171386050150400000000) |
| (29) | (884176199373970195454361600000000) |
| (30) | (26525285981219105863630848000000000) |
调和与超调和数表
调和与超调和数在级数理论等方面有重要应用。以下是部分调和与超调和数的数表：
| (n) | (H_n) | (H_n^2) | (H_n^3) | (H_n^4) |
| — | — | — | — | — |
| (0) | (0) | (0) | (0) | (0) |
| (1) | (1) | (1) | (1) | (1) |
| (2) | (\frac{3}{2}) | (\frac{5}{2}) | (\frac{7}{2}) | (\frac{9}{2}) |
| (3) | (\frac{11}{6}) | (\frac{13}{3}) | (\frac{47}{6}) | (\frac{37}{3}) |
| (4) | (\frac{25}{12}) | (\frac{77}{12}) | (\frac{57}{4}) | (\frac{319}{12}) |
| (5) | (\frac{137}{60}) | (\frac{87}{10}) | (\frac{459}{20}) | (\frac{743}{15}) |
| (6) | (\frac{49}{20}) | (\frac{223}{20}) | (\frac{341}{10}) | (\frac{2509}{30}) |
| (7) | (\frac{363}{140}) | (\frac{481}{35}) | (\frac{3349}{70}) | (\frac{2761}{21}) |
| (8) | (\frac{761}{280}) | (\frac{4609}{280}) | (\frac{3601}{56}) | (\frac{32891}{168}) |
| (9) | (\frac{7129}{2520}) | (\frac{4861}{252}) | (\frac{42131}{504}) | (\frac{35201}{126}) |
| (10) | (\frac{7381}{2520}) | (\frac{55991}{2520}) | (\frac{44441}{420}) | (\frac{485333}{1260}) |
幂等数表
幂等数在抽象代数等领域有其独特意义。幂等数（A000248）的数表如下：
| (n) | (\imath_n) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (3) |
| (3) | (10) |
| (4) | (41) |
| (5) | (196) |
| (6) | (1057) |
| (7) | (6322) |
| (8) | (41393) |
| (9) | (293608) |
| (10) | (2237921) |
| (11) | (18210094) |
| (12) | (157329097) |
| (13) | (1436630092) |
| (14) | (13810863809) |
| (15) | (139305550066) |
| (16) | (1469959371233) |
| (17) | (16184586405328) |
| (18) | (185504221191745) |
| (19) | (2208841954063318) |
| (20) | (27272621155678841) |
| (21) | (348586218389733556) |
| (22) | (4605223387997411873) |
| (23) | (62797451641106266330) |
| (24) | (882730631284319415505) |
| (25) | (12776077318891628112376) |
| (26) | (190185523485851040093857) |
| (27) | (2908909247751545392493182) |
| (28) | (45671882246215264120864553) |
| (29) | (735452644411097903203941148) |
| (30) | (12136505435201514536093218561) |
对合数、受限贝尔数、受限阶乘数、关联贝尔数、关联阶乘数表
-
对合数（A000085） ：在组合数学中与集合的对合等概念相关。
| (n) | (I_n) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (2) |
| (3) | (4) |
| (4) | (10) |
| (5) | (26) |
| (6) | (76) |
| (7) | (232) |
| (8) | (764) |
| (9) | (2620) |
| (10) | (9496) |
| (11) | (35696) |
| (12) | (140152) |
| (13) | (568504) |
| (14) | (2390480) |
| (15) | (10349536) |
| (16) | (46206736) |
| (17) | (211799312) |
| (18) | (997313824) |
| (19) | (4809701440) |
| (20) | (23758664096) |
| (21) | (119952692896) |
| (22) | (618884638912) |
| (23) | (3257843882624) |
| (24) | (17492190577600) |
| (25) | (95680443760576) |
| (26) | (532985208200576) |
| (27) | (3020676745975552) |
| (28) | (17411277367391104) |
| (29) | (101990226254706560) | -
2 - 受限贝尔数（A006505） ：在受限条件下的集合划分问题中有应用。
| (n) | (B_{n,\leq2}) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (2) |
| (3) | (5) |
| (4) | (14) |
| (5) | (46) |
| (6) | (166) |
| (7) | (652) |
| (8) | (2780) |
| (9) | (12644) |
| (10) | (61136) |
| (11) | (312676) |
| (12) | (1680592) |
| (13) | (9467680) |
| (14) | (55704104) |
| (15) | (341185496) |
| (16) | (2170853456) |
| (17) | (14314313872) |
| (18) | (97620050080) |
| (19) | (687418278544) |
| (20) | (4989946902176) |
| (21) | (37286121988256) |
| (22) | (286432845428192) |
| (23) | (2259405263572480) |
| (24) | (18280749571449664) |
| (25) | (151561941235370176) |
| (26) | (1286402259593355776) |
| (27) | (11168256342434121152) |
| (28) | (99099358725069658880) |
| (29) | (898070590439513534464) |
| (30) | (8306264068494786829696) | -
2 - 受限阶乘数（A057693） ：在受限排列问题中有其作用。
| (n) | (A_{n,\leq2}) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (2) |
| (3) | (6) |
| (4) | (18) |
| (5) | (66) |
| (6) | (276) |
| (7) | (1212) |
| (8) | (5916) |
| (9) | (31068) |
| (10) | (171576) |
| (11) | (1014696) |
| (12) | (6319512) |
| (13) | (41143896) |
| (14) | (281590128) |
| (15) | (2007755856) |
| (16) | (14871825936) |
| (17) | (114577550352) |
| (18) | (913508184096) |
| (19) | (7526682826848) |
| (20) | (64068860545056) |
| (21) | (561735627038496) |
| (22) | (5068388485760832) |
| (23) | (47026385852423616) |
| (24) | (447837548306401728) |
| (25) | (4374221252904547776) |
| (26) | (43785991472018760576) |
| (27) | (448610150446698125952) |
| (28) | (4701535239730197200256) |
| (29) | (50364829005083927722368) |
| (30) | (550980793119978524802816) | -
2 - 关联贝尔数（A000296） ：在特定关联条件下的集合划分计数。
| (n) | (B_{n,\geq2}) |
| — | — |
| (0) | (1) |
| (1) | (0) |
| (2) | (1) |
| (3) | (1) |
| (4) | (4) |
| (5) | (11) |
| (6) | (41) |
| (7) | (162) |
| (8) | (715) |
| (9) | (3425) |
| (10) | (17722) |
| (11) | (98253) |
| (12) | (580317) |
| (13) | (3633280) |
| (14) | (24011157) |
| (15) | (166888165) |
| (16) | (1216070380) |
| (17) | (9264071767) |
| (18) | (73600798037) |
| (19) | (608476008122) |
| (20) | (5224266196935) |
| (21) | (46499892038437) |
| (22) | (428369924118314) |
| (23) | (4078345814329009) |
| (24) | (40073660040755337) |
| (25) | (405885209254049952) |
| (26) | (4232705122975949401) |
| (27) | (45398541400642806873) |
| (28) | (500318506535417182516) |
| (29) | (5660220898064517469939) |
| (30) | (65679581040795757721233) | -
2 - 关联阶乘数（A000166） ：在关联排列问题中有应用。
| (n) | (A_{n,\geq2}) |
| — | — |
| (0) | (1) |
| (1) | (0) |
| (2) | (1) |
| (3) | (2) |
| (4) | (9) |
| (5) | (44) |
| (6) | (265) |
| (7) | (1854) |
| (8) | (14833) |
| (9) | (133496) |
| (10) | (1334961) |
| (11) | (14684570) |
| (12) | (176214841) |
| (13) | (2290792932) |
| (14) | (32071101049) |
| (15) | (481066515734) |
| (16) | (7697064251745) |
| (17) | (130850092279664) |
| (18) | (2355301661033953) |
| (19) | (44750731559645106) |
| (20) | (895014631192902121) |
| (21) | (18795307255050944540) |
| (22) | (413496759611120779881) |
| (23) | (9510425471055777937262) |
| (24) | (228250211305338670494289) |
| (25) | (5706255282633466762357224) |
| (26) | (148362637348470135821287825) |
| (27) | (4005791208408693667174771274) |
| (28) | (112162153835443422680893595673) |
| (29) | (3252702461227859257745914274516) |
| (30) | (97581073836835777732377428235481) |
拉赫数与超阶乘数表
-
拉赫数（A105278） ：在组合数学的某些变换中有应用。
| (n \diagdown k) | (k = 1) | (k = 2) | (k = 3) | (k = 4) | (k = 5) | (k = 6) | (k = 7) | (k = 8) |
| — | — | — | — | — | — | — | — | — |
| (n = 1) | (1) | | | | | | | |
| (n = 2) | (2) | (1) | | | | | | |
| (n = 3) | (6) | (6) | (1) | | | | | |
| (n = 4) | (24) | (36) | (12) | (1) | | | | |
| (n = 5) | (120) | (240) | (120) | (20) | (1) | | | |
| (n = 6) | (720) | (1800) | (1200) | (300) | (30) | (1) | | |
| (n = 7) | (5040) | (15120) | (12600) | (4200) | (630) | (42) | (1) | |
| (n = 8) | (40320) | (141120) | (141120) | (58800) | (11760) | (1176) | (56) | (1) |
| (n = 9) | (362880) | (1451520) | (1693440) | (846720) | (211680) | (28224) | (2016) | (72) | -
超阶乘数（A000178） ：在高阶组合计数等方面有意义。
| (n) | (sf(n)) |
| — | — |
| (0) | (1) |
| (1) | (1) |
| (2) | (2) |
| (3) | (12) |
| (4) | (288) |
| (5) | (34560) |
| (6) | (24883200) |
| (7) | (125411328000) |
| (8) | (5056584744960000) |
| (9) | (18349334722510848000000) |
| (10) | (66586065841047365222400000000) |
数表应用总结与拓展
这些数表涵盖了组合数学、数论、数学分析等多个领域的重要概念和工具。在实际应用中，我们可以根据问题的具体特点，选择合适的数表进行计算和分析。以下是一个更详细的流程图，展示了在解决复杂数学问题时，如何综合运用这些数表：
graph TD;
A[复杂数学问题] --> B{问题领域};
B -->|组合数学| C{组合类型};
C -->|集合划分| D[贝尔数、斯特林数];
C -->|排列模式| E[欧拉数];
C -->|其他组合场景| F[其他相关数表];
B -->|数论| G{数论问题类型};
G -->|整除性问题| H[伯努利数、柯西数等];
G -->|方程求解| I[根据具体方程选择数表];
B -->|数学分析| J{分析问题类型};
J -->|级数问题| K[调和与超调和数等];
J -->|函数逼近| L[根据函数性质选择数表];
D --> M[根据数表计算中间结果];
E --> M;
F --> M;
H --> M;
I --> M;
K --> M;
L --> M;
M --> N[综合分析结果];
N --> O[得出最终解决方案];
通过对这些数表的深入研究和应用，我们可以更好地理解数学的内在结构和规律，为解决各种实际问题提供有力的支持。同时，这些数表之间的关系和相互作用也为数学研究提供了广阔的探索空间，值得我们不断去挖掘和发现。
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