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Sequences from Comtet's Advanced Combinatorics

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Keywords: Comtet, Advanced Combinatorics, Analyse Combinatoire, Integer Sequences, Concordance

  • For a long time I (njas) have had the idea of making a series of concordances which would list the integer sequences to be found in certain classical books (Riordan, Stanley, Harary and Palmer, Knuth, Graham-Knuth-Patashnik, etc.). Here is the second of these, a concordance to
    Louis Comtet's Advanced Combinatorics, Reidel, 1974
  • The idea is that when you are reading one of these books, these files will give pointers to the On-Line Encyclopedia of Integer Sequences whenever an interesting sequence is mentioned. This will enable you to see the terms of the sequence, recurrences, formulae, other references, links, most recent progress, etc.
  • Furthermore, the preparation of these concordances will supply additional sequences for the database, and additional references for existing sequences.
  • The French edition of this book, Analyse Combinatoire, was published in 1970 by Presses Universitaires de France, Paris, in two wonderful pocket-sized paperbacks.
  • The revised and enlarged English edition, Advanced Combinatorics, was published in 1974 by D. Reidel, Dordrecht, Holland. All page references here refer to the English edition.
  • In case I have missed any sequences, please either edit this page, or send suggestions to Neil Sloane at this address:
  • Entries marked (****) indicate sequences that may need to be added to the database. Help will be welcomed.
  • For the current list of these concordances, see here.

Louis Comtet, Advanced Combinatorics

Chapter 1: Vocabulary of Combinatorial Analysis

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

Chapter 2: Partitions of Integers

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(1) A008288, A113139 and #18(2) A009766, A033184
Page 122, #19(2), I(n) is A059293; #19(3(1)), I(n) is A000330. #19(3(2)) is A321986. #19(3(3)) is A321988
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

Chapter 3: Identities and Expansions

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 158, #7, Z_{n,k}(2) is A185296.
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

Chapter 4: Sieve Formulas

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

Chapter 5: Stirling Numbers

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}: These are fractions of the form A048994(r,r-n)/binomial(r-1,n).
Page 228, #19: A001147
Page 228, #20, a_m: A000670
Page 229, #25, t_n: A000798

Chapter 6: Permutations

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

Chapter 7: Examples of Inequalities and Estimates

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

Fundamental Numerical Tables

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

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