cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A059825 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^8 *product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 9, 44, 164, 485, 1278, 2949, 6382, 12661, 24101, 43063, 74932, 124041, 201597, 315048, 485627, 724514, 1071104, 1539099, 2197385, 3062512, 4246873, 5765303, 7804391, 10359671, 13728320, 17882076, 23264374, 29792631, 38154696
Offset: 0

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Links

  • G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Crossrefs

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Programs

  • Maple
    Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=8

A059819 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^2 *Product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 18, 19, 28, 30, 40, 39, 57, 51, 68, 68, 86, 77, 107, 91, 123, 114, 138, 121, 172, 140, 178, 166, 205, 171, 240, 189, 251, 224, 266, 230, 322, 245, 314, 286, 356, 283, 396, 303, 403, 361, 416, 343, 497, 368, 479, 424, 515, 407
Offset: 0

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Links

Crossrefs

Cf. A000005 (k=1), here (k=2), A059820 (k=3), ..., A059825 (k=8).
Cf. A000203, A055507.

Programs

  • Maple
    Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=2

Formula

a(n) = (sigma(n)+tau(n)+Sum_{k=0..n} tau(k)*tau(n-k))/2.
G.f.: (F(x)+G(x)^2)/2, where F(x) = Sum_{k>0} (k+1)*x^k/(1-x^k) and G(x) = Sum_{k>0} x^k/(1-x^k). - Vladeta Jovovic, Feb 12 2004

A059823 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^6 *product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 7, 27, 83, 202, 455, 889, 1682, 2892, 4894, 7694, 12090, 17822, 26411, 37206, 52730, 71447, 97984, 128714, 171421, 220064, 285963, 359204, 458506, 565347, 708665, 862163, 1064302, 1276474, 1558090, 1845874, 2226044, 2614188
Offset: 0

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Links

  • G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Crossrefs

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Programs

  • Maple
    Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=6

A059821 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^4 *product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 5, 14, 34, 64, 121, 190, 311, 446, 666, 887, 1266, 1599, 2169, 2679, 3504, 4178, 5383, 6253, 7858, 9060, 11114, 12560, 15390, 17076, 20512, 22788, 26993, 29494, 34988, 37750, 44213, 47857, 55281, 59196, 68810, 72754, 83518, 88947
Offset: 0

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Links

  • G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Crossrefs

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Programs

  • Maple
    Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=4

A059822 Expansion of series related to Liouville's Last Theorem: g.f. sum_{t>0} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^5 *product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 6, 20, 55, 119, 246, 435, 766, 1211, 1926, 2807, 4193, 5766, 8161, 10821, 14711, 18820, 24925, 31009, 39984, 48895, 61609, 73844, 91905, 108264, 132400, 154641, 186462, 214772, 257118, 292749, 346430, 392499, 459424, 515579
Offset: 0

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Links

  • G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Crossrefs

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Programs

  • Maple
    Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=5

A059824 Expansion of series related to Liouville's Last Theorem: g.f. Sum_{t>=1} (-1)^(t+1) *x^(t*(t+1)/2) / ( (1-x^t)^7 * Product_{i=1..t} (1-x^i) ).

Original entry on oeis.org

0, 1, 8, 35, 119, 321, 784, 1672, 3389, 6280, 11285, 18971, 31383, 49162, 76322, 113494, 167785, 239086, 340355, 468636, 646058, 865724, 1161936, 1520105, 1997015, 2559758, 3297599, 4157592, 5266644, 6537922, 8168293, 10003615
Offset: 0

Views

Author

N. J. A. Sloane, Feb 24 2001

Keywords

Links

  • G. E. Andrews, Some debts I owe, Séminaire Lotharingien de Combinatoire, Paper B42a, Issue 42, 2000; see (7.4).

Crossrefs

Cf. A000005 (k=1), A059819 (k=2), A059820 (k=3), ..., A059825 (k=8).

Programs

  • Maple
    Mk := proc(k) -1*add( (-1)^n*q^(n*(n+1)/2)/(1-q^n)^k/mul(1-q^i,i=1..n), n=1..101): end; # with k=7
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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