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 1-8 of 8 results.

A028377 Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).

Original entry on oeis.org

1, 1, 3, 9, 19, 46, 100, 218, 460, 965, 1975, 3993, 7975, 15712, 30650, 59150, 113093, 214300, 402812, 751165, 1390714, 2557004, 4670770, 8479232, 15302657, 27462424, 49021252, 87057783, 153850769, 270614429, 473850031, 826125184, 1434286323, 2480145226
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Convolved with aerated A000294: [1, 0, 2, 0, 4, 0, 10, 0, 26, ...] = A000294. - Gary W. Adamson, Jun 13 2009
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(n+1)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Links

Crossrefs

Cf. A000294. - Gary W. Adamson, Jun 13 2009
Cf. A027999, A258341, A258342, A258343, A258344, A258345, A258346.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i*(i+1)/2, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 03 2013
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[i*(i+1)/2, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

Formula

a(n) ~ 7^(1/8) * exp(2 * 7^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) + 3^(3/2) * 5^(1/2) * Zeta(3) * n^(1/2) / (2 * 7^(1/2) * Pi^2) - 3^(13/4) * 5^(5/4) * Zeta(3)^2 * n^(1/4) / (4 * 7^(5/4) * Pi^5) + 2025 * Zeta(3)^3 / (98*Pi^8)) / (2^(49/24) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(d+1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 28 2018

A258349 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 1, 3, 7, 13, 28, 52, 107, 203, 396, 741, 1409, 2596, 4813, 8777, 15972, 28737, 51553, 91644, 162288, 285377, 499653, 869758, 1507615, 2599974, 4465606, 7635607, 13005252, 22061424, 37287395, 62788012, 105365891, 176211393, 293741195, 488101711, 808604106
Offset: 0

Views

Author

Vaclav Kotesovec, May 27 2015

Keywords

Links

Crossrefs

Cf. A000294, A023871, A027999, A258347, A258348, A258350, A258351, A258352.

Programs

  • Mathematica
    nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)/2),{k,1,nmax}],{x,0,nmax}],x]
  • SageMath
    # uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n,2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

Formula

a(n) ~ 1 / (2^(155/96) * 15^(11/96) * Pi^(1/24) * n^(59/96)) * exp(-Zeta'(-1)/2 - Zeta(3) / (8*Pi^2) - 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 / (2^(7/4) * Pi^5) * n^(1/4) - sqrt(15/2) * Zeta(3) / Pi^2 * sqrt(n) + 2^(7/4)*Pi / (3*15^(1/4)) * n^(3/4)), where Zeta(3) = A002117, Zeta'(-1) = A084448 = 1/12 - log(A074962).
G.f.: exp(Sum_{k>=1} (sigma_3(k) - sigma_2(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 22 2018

A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

Original entry on oeis.org

1, 2, 7, 24, 65, 184, 487, 1254, 3145, 7706, 18480, 43490, 100692, 229472, 515802, 1144416, 2508948, 5439642, 11671859, 24801738, 52221911, 109013538, 225718717, 463769652, 945915199, 1915895576, 3854803572, 7706786958, 15314564282, 30255672820, 59440488874
Offset: 0

Views

Author

Vaclav Kotesovec, May 27 2015

Keywords

Links

Crossrefs

Cf. A027998, A028377, A027999, A258342, A258343, A258344, A258345, A258346, A258347.

Programs

  • Mathematica
    nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) + 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

Original entry on oeis.org

1, 0, 2, 6, 13, 32, 69, 160, 344, 760, 1601, 3384, 7022, 14434, 29361, 59140, 118089, 233754, 459293, 895382, 1733904, 3334914, 6374654, 12111632, 22881777, 42993244, 80362496, 149464404, 276657082, 509740278, 935046158, 1707916988, 3106810873, 5629121054
Offset: 0

Views

Author

Vaclav Kotesovec, May 27 2015

Keywords

Links

Crossrefs

Cf. A027998, A028377, A027999, A258341, A258342, A258343, A258345, A258346, A258348.

Programs

  • Mathematica
    nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ 7^(1/8) / (2^(43/24) * 15^(1/8) * n^(5/8)) * exp(-2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) - 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A292386 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)/2).

Original entry on oeis.org

1, -1, -3, -3, -1, 10, 20, 36, 28, -11, -103, -245, -397, -448, -214, 464, 1817, 3680, 5660, 6473, 4362, -3232, -18428, -41946, -70589, -94890, -96996, -49673, 78907, 317995, 673299, 1105044, 1491333, 1605102, 1094914, -479358, -3561322, -8404118, -14781724, -21595744, -26450603, -25329527
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 15 2017

Keywords

Comments

Convolution inverse of A000294 (Euler transform of the triangular numbers).

Crossrefs

Cf. A000294, A027999, A028377.

Programs

  • Mathematica
    nmax = 41; CoefficientList[Series[Product[(1 - x^k)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
  • SageMath
    # uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

Formula

G.f.: Product_{k>=1} (1 - x^k)^(k*(k+1)/2).

A294780 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 2, 6, 14, 32, 74, 166, 370, 810, 1736, 3682, 7718, 15976, 32754, 66508, 133794, 266948, 528424, 1038178, 2025456, 3925360, 7559298, 14470162, 27540598, 52130440, 98159832, 183905636, 342896254, 636384748, 1175823512, 2163221030, 3963353706, 7232529308
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2017

Keywords

Comments

Convolution of A027999 and A258349.

Links

Crossrefs

Cf. A027999, A156616, A206622, A258349, A294779.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k*(k-1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2*Pi * n^(3/4) / 3 - 7*Zeta(3) * sqrt(n) / (2*Pi^2) - 49*Zeta(3)^2 * n^(1/4) / (4*Pi^5) - 22411 * Zeta(3)^3 / (392*Pi^8) - Zeta(3) / (8*Pi^2) - 1/24) * sqrt(A) / (2^(23/12) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A294777 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 3, 0, 6, 3, 10, 9, 15, 28, 24, 60, 47, 126, 99, 227, 225, 414, 498, 717, 1044, 1301, 2082, 2364, 3984, 4482, 7353, 8513, 13287, 16317, 23698, 30789, 42081, 57499, 74763, 105276, 133273, 190155, 238122, 338291, 425775, 596142, 759651, 1041498
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 08 2017

Keywords

Crossrefs

Cf. A027998, A027999, A263140, A294749.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1+x^(2*k-1))^(k*(k-1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*14^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi*5^(1/4) * n^(1/4) / (2^(17/4) * 3^(3/4) * 7^(1/4))) * 7^(1/8) / (2^(19/8) * 15^(1/8) * n^(5/8)).

A298850 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^(k*(k+1)/2).

Original entry on oeis.org

1, 1, 0, 3, 3, 0, 9, 9, 0, 19, 29, 10, 33, 63, 30, 66, 156, 90, 110, 300, 235, 276, 561, 465, 558, 1083, 1065, 1154, 1877, 1983, 2295, 3834, 3879, 3861, 6858, 7452, 7561, 12613, 13252, 13057, 22161, 25569, 24582, 35985, 44193, 44970, 63495, 79105, 77143, 104046, 134820, 138759, 182511, 222600
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2018

Keywords

Crossrefs

Cf. A000217, A024940, A027999, A028377, A291649, A298730.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A000217(k))^A000217(k).
Showing 1-8 of 8 results.

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

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