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Keywords: Comtet, Advanced Combinatorics, Analyse Combinatoire, Integer Sequences, Concordance
Page 6: n! A000142
Page 12 Fig. 1 - see p. 306 for a larger table
Page 45 Fibonacci numbers F_n A000045
Page 46 G_n Apart from the initial 0, this is A000204
Page 48, Section 1.14 (I): [14a] Bernoulli numbers B_n: A027641/A027642; B_{2n}: A000367/A002445; Bernoulli polynomials B_n(x): A053382/A053383
[14b] Euler numbers E_n: A000364; Euler polynomials E_n(x): A059341/A059342 etc.
Page 49 Bernoulli numbers B_n: A027641/A027642; B_{2n}: A000367/A002445
Page 49 Euler numbers E_n: A000364.
Page 49 Genocci numbers G_n: A001469
Page 49, Section 1.14 (II): [14i] Chebysheff polynomials of first kind: the database contains many related sequences - see index entries for sequences related to Chebyshev polynomials.
Page 50 [14j] Chebysheff polynomials of second kind: the database contains many related sequences - see index entries for sequences related to Chebyshev polynomials.
[14l] Legendre polynomials: A008316
[14n] Hermite polynomials: A059343
[14o] Laguerre polynomials: A021009 (?)
Page 50, Section 1.14 (III): [14p] Stirling numbers of first kind s(n,k): A008275; [14q] Stirling numbers of second kind S(n,k): A008277;
Page 51, Section 1.14 (IV): [14t] Eulerian numbers A(n,k): A008292;
Page 53 Catalan numbers C_n A000108
Page 55 b_n A001190
Page 57 Schroeder's second problem c_n A001003
Page 60 d_n A001035
Page 60 d*_n A000112
Page 60 D(n,k) triangle is A008285; columns and diagonals give A055531, A055532, A000142, A055533, A055534
Page 63 Theorem D A000272
Page 72 #1 A000217, A050534, A055503
Page 72 #2(1) A000124
Page 72 #2(2) A000127, A006261, A008859, A008860, A008861, A008862, A008863
Page 73 #3 A014206, A055504
Page 73 #4 A046127, A059173, A059174, A059214, A059250
Page 74 #7(1) A005044
Page 74 #7(2) A002623
Page 74 #8(1) A000332, A005701
Page 74 #8(2) A006522, A055503
Page 74 #8(3) A000108
Page 75 #8(5) (1/6!)*n*(n-1)*(n-2)*(n^3+18*n^2+43*n+60) is not always an integer
Page 75 #9 A000522
Page 78 Triangle of trinomial coefficients: A027907. Columns and diagonals give A000217, A005581, A005712, A000574, A005714, A005715, A005716, A002426, A005717, A014531, A014532, A014533. See also A035000, A014531.
Page 78 Triangle of quadrinomial coefficients: A008287. Columns and diagonals give A000217, A000292, A005718, A005719, A005720, A001919, A005190, A005721, A005723, A005724, A005725, A005726
Page 81, #21 (2) Triangle of Delannoy numbers: A008288. Rows, diagonals give A001844, A001845, A001846, A001847, A001848, A001849, A001850.
Page 81, #21 (3) P_n(3) is A001850
Page 81, #21 (4) q_n: A006318; c_n: A001003
Page 83, #25, Leibniz's triangle: A003506. See also A002457, A007622, A046200, A046201, A046202, A046203, A046204, A046205, A046206, A046207, A046208, A046212
Page 84, #25, c(n) A003319
Page 87, #32, S' A005647
Page 88, #36, tanh(x): A000182 and A002430/A036279
Page 88, #36, tan(x): A000182 and A002430/A036279
Page 88, #36, cot(x): A002431/A036278
Page 88, #36, arcsin(x): A055786/A002595
Page 88, #36, log(cos(x)): A046990/A046991
Page 88, #36, log(sin(x)/x): A046988/A046989
Page 89, #36, zeta(2n): A002432
Page 89, #36, Bernoulli numbers B_n: A027641/A027642; B_{2n}: A000367/A002445
Page 89, #37, Euler numbers E_n: A000364
Page 89, #37, beta(n): A053005
Page 91, #42, filter bases: A059301
Page 91, #43, the triangle of idempotent numbers binomial(n,k)*k^(n-k) appears in four versions, A059297, A059298, A059299 and A059300. The diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums i(n) are A000248.
Page 93, #47: see A003016, A003015, A059233
Page 94, partitions, p(n): A000041
Page 94, partitions of n into m parts, P(n,m): A008284 (read by rows, from left to right), A058398 (read by rows, right to left)
Page 96, partitions of n into at most m parts, p(n,m): A008284 (read by antidiagonals downwards), A058398 (read by antidiagonals upwards)
Page 99, partitions into distinct parts, q(n): A000009
Page 104, coefficients in expansion of [5g], from Euler's Pentagonal Theorem: A010815 (essentially expansion of Dedekind eta function)
Page 106, [5n]: coefficients in expansion of theta_3, from Jacobi Triple Product Identity: A000122
Pages 107-108: coefficients in expansion of Rogers-Ramanujan identities: A003114 and A003106
Page 109, [6c] D(n; 1,2): A008619; [6d] D_(1,2,3): A001399
Page 110, D(n) = D(n; 1,2,3): A001399
Page 112, D(n) = D(n; 1,2,4): A008642
Page 113, D(n; 1,2,3): A001399
Page 113, D(n; 1,2,4,10): A001304 (with terms repeated)
Page 114, D(n; 3,5,7): A008677
Page 115, partitions of n into m different parts, Q(n,m): A008289 (read by columns). Rows of table give A001399, A001400, A001401, etc.
Page 115, p*(n): A002865. Second differences of p(n): A053445. Third, fourth and fifth differences (which are only >= 0 for n sufficiently large): A072380, A081094, A081095
Page 116, #4. P(n,2) (or Q(n,2)) = A004526, A008619; P(n,3) (or Q(n,3)) = A001399; P(n,4) (or Q(n,4)) = A001400, A026810
Page 117, p_1(n) = partitions into distinct parts, q(n): A000009
Page 118, #10, D(n) = binary weight: A000120
Page 118, #11: there are many sequences of q-binomial coefficients in the database - see the index
Page 118, #12, omega(n): A001222
Page 120, #15: D(n;1,2,5): A000115; D(n;1,2,7): A025764; D(n;1,3,5): A008672;
D(n;1,3,7): A025768; D(n;1,5,7): A025777; D(n;1,2,3,5): A008669;
P(n,2): A004526; P(n,3): A001399; P(n,4): A001400 and A026810, see also A059290, A059291
Page 121, #17, A059292
Page 121, the triangles of numbers in #18 ? (****)
Page 122, #19(2), I(n) is A059293; #19(3(1)), I(n) is A000330. What about I(n) in (3(2)) and (3(3))? (****)
Page 123, #20, f(n) is A001192
Page 123, #21, s(n) is A000571
Page 124, #25: Q(r,3) is A002817, Q(4,r) is A001496
Page 125, #25: a_n = Q(n,2) is A000681; A_n is A005650; b_n (the n=3 term is wrong) is A001500
Page 126, #27, perfect partitions: A002033
Page 126, #28, A(n) is A005651
Page 135, Stirling numbers of second kind: A008277; Lah numbers: A008297; Stirling numbers of first kind: A008275; idempotent numbers: A059297, A059298, A059299 and A059300
Page 139, Triangle b(n,k) is A008296. Diagonals give A000142, A045406, A000217, A059302. Row sums give A005727.
Page 148, table is A008826; columns and diagonals give A008827, A006472, A059359
Page 155, sum of first n k-th powers, for k = 1 ... 8: A000217, A000330, A000537, A000538, A000539, A000540, A000541, A000542
Page 156, #2, Triangle of Lah numbers: A008297. See also A007318, A048786. Row sums of unsigned triangle form A000262(n). A002868 gives maximal element (in magnitude) in each row.
Page 159, #10, Triangle is A008298. Diagonals give A038048, A059356, A059357.
Page 161, #16, (1)-(3) Moebius function: A008683
Page 162, #16 (4) d(n): A000005; (5) phi(n): A000010; (6) expansions of the four identities give A048272, A000203, A002129, A017665/A017666; (7), r(n): A004018
Page 163, #19, trinomial coefficients a_n: A002426 (see also A027907)
Page 167, #27, a(m,s) gives A059366; main diagonal is A001757
Page 168, #30, 3rd formula gives A002457.
Page 169, #31, last formula: A002593
Page 170, #33, C(m,k): A059368. First column is A001147.
Page 171, #33, c_m: A059367
Page 171, #34, A(n,k): A059369; a(n,k): A059370. Diagonals give A000142, A059371, A059372, A059373.
Page 173, #39: A000312
Page 174, #42: A006480
Page 175, #44, a(n): A003262
Page 180, derangements d(n): A000166
Page 182, derangements d(n): A000166
Page 183, K(n): A000186; l(n): A000315
Page 184, mu(n): A000179, mu*(n): A059375
Page 185, mu(n): A000179
Page 193, phi(n): A000010
Page 199, #2: A053818
Page 199, #3, the Jordan function J_k(n), a generalization of the phi function: array gives A059379 and A059380; rows (for k = 1,2,3,4,5) give A000010, A007434, A059376, A059377, A059378; columns give A000225, A024023, A020522, A024049, A059387, A059409, A059410
Page 203, #17, product of phi(i): A00108; product J_k(i) for i = 2,3,4,5: A059381, A059382, A059383, A059384
Page 204: Stirling numbers of first kind s(n,k): A008275; Stirling numbers of second kind S(n,k): A008277;
Page 210: Bell numbers omega(n): A000110
Page 212, the table is Aitken's array, A011971. Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc.; A011968, A011969
Page 212: Stirling numbers of first kind s(n,k): A008275
Page 217: s(n,2), s(n,3), s(n,4): A000254, A000399, A000454
Page 222, #7, triangle of associated Stirling numbers of second kind: A008299. Rows give A000247, A000478, A058844
Page 224, #10, s_n: A000311
Page 226, #13, table of exponential generating functions, last column: A000110, A003724, A005046, A024429, A024430, A003712, A059385, A003709, A059386
Page 227, #16: S(n,n-1): A000217; S(n,n-2): A001296; S(n,n-3): A001297; s(n,n-1): A000217; s(n,n-2): A000914; s(n,n-3): A001303; s(n,n-4): A000915
Page 227, #18, generalized Bernoulli numbers B_n^{r}: I believe several of these sequences are also in the database (check!) (****)
Page 228, #19: A001147
Page 228, #20, a_m: A000670
Page 229, #25, t_n: A000798
Page 236, triangle P(n,k): A008300; diagonals give A000142, A001499, A001501, A058527
Page 240, b(n,k): A008302. Diagonals give A000707, A001892, A001893, A001894, A005283, A005284, A005285, A005286, A005287, A005288.
Page 243, Eulerian numbers A(n,k): A008292. Columns 2 through 8: A000295, A000460, A000498, A000505, A000514, A001243, A001244.
Page 255, #2, b(n,3): A005286; b(n,4): A005287
Page 256, #7, triangle of d(n,k): A008306. Rows give A000142, A000276, A000483.
Page 257, #9, array T(n,k) gives A008307. Rows give A056595, (more sequences needed!); columns give A000085, A001470, etc (more cross-references needed!). (****)
Page 258, #10, triangle F(n,k) is A059418; diagonals give A001710, A006595.
Pages 258-260, #11, A_n is A000111. Triangle of T(n,k) is A059419 (and A008308); diagonals give A000182, A024283, A059420, A059421, A007290. Row sums give A006229.
Page 260, #10, cont. Triangle of t(n,k) is A008309 (and A049218); diagonals give A007290(n)=-t(n,[ (n-1)/2 ]); A010050(n)=(-1)^n*t(2n+1,1); A049034(n)=(-1)^n*t(2n+2,1); A049214(n)=(-1)^n*t(2n+3,2); A049215(n)=(-1)^n*t(2n+4,2); A049216(n)=(-1)^n*t(2n+5,3); A049217(n)=(-1)^n*t(2n+6,3).
Page 260, #11: a_n is A002135, a'_n is A059422, p_n is A059423, q_n is A059424
Page 261, #13, P(n,k) gives A008970 and A059427. Diagonals give A001250, A059428, A028399. A_n is A000111.
Page 262, #14, triangle is A059438. Diagonals give A003319, A059439, A059440, A055998.
Page 263, #18, P_n^{3} is A001399.
Page 264, #19, triangle of g_{n,k} (which should be preceded by a column of 1's): A008406.
Page 267, #22, a(n): A000560
Page 267, #23, c_q: A001163/A001164
Page 273, s(n): A000372, A003182, A007153
Page 276, g_n: A001205
Page 279, triangle of G(n,r): A059441. Diagonals give A001205, A002829, A005815
Page 288, rho(p,q): A059422; rho(p,2) is A000791 (many entries in this table have been improved).
Page 291, first two rows of table give A001197, A001198
Page 292, #8, s(n,2): A016269, s(n,3): A047707, s(n,4): A051112. See also A051119.
Page 293, #11, A(n) = A055505/A055535.
Page 294, #13, a_n: A006232/A006233, b_n: A002657/A002790
Page 294, #14, A(n): A000990
Page 294, #15, b_p: A005649
Page 295, #16, f(n) = A003319
Page 295, #20, associated Stirling numbers of second kind: A008306; d(n,k): A008306.
Page 301-302, #35, number of groups of order n, g(n): A000001
Page 303, #40, C_2(n,k): A059443; C_2(n): A002718; C_2(n,3): A003462.
Page 303, #42, g(n): A056642 (the version Comtet gives is A001199, but this is believed to be incorrect); g*(n): A001200
Page 304, #43, s(n): A001201; s*(n): A051391
Page 305, n!: A000142; exponent of 2 in n!: A011371
Page 306, Pascal's triangle of binomial coefficients: A007318
Page 307, p(n) = partition numbers = A000041; triangle P(n,m): A008284 and A058398
Page 310, Stirling numbers of first kind s(n,k): A008275; Stirling numbers of second kind S(n,k): A008277; omega(n): A000110