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.

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A294779 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k-1)))^(k*(k-1)/2).

Original entry on oeis.org

1, 0, 0, 2, 0, 6, 2, 12, 12, 22, 42, 42, 114, 102, 264, 280, 564, 744, 1186, 1866, 2538, 4380, 5598, 9732, 12602, 20898, 28374, 44048, 63000, 92190, 137012, 192864, 291588, 403668, 609072, 843228, 1253978, 1752150, 2555058, 3611380, 5168778, 7371324, 10400908
Offset: 0

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Author

Vaclav Kotesovec, Nov 08 2017

Keywords

Comments

Convolution of A294777 and A294778.

Crossrefs

Cf. A206622, A292037, A294755, A294777, A294778, A294780.

Programs

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

Formula

a(n) ~ exp(Pi * 2^(1/4) * n^(3/4)/3 - Pi*n^(1/4) / 2^(17/4) + 3*Zeta(3) / (32*Pi^2)) / (2^(37/16) * n^(5/8)).

A302238 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^prime(k).

Original entry on oeis.org

1, 4, 14, 46, 136, 382, 1022, 2626, 6530, 15784, 37218, 85842, 194146, 431358, 943038, 2031454, 4316884, 9058662, 18787730, 38542526, 78264298, 157403290, 313712482, 619919350, 1215125262, 2363570168, 4563951858, 8751621598, 16670498062, 31553539214, 59361428202
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Comments

Convolution of the sequences A030009 and A061152.

Links

Crossrefs

Cf. A000040, A015128, A030009, A061152, A156616, A206622, A206623, A206624, A260916, A261386, A261452, A261519, A261520, A301554, A301555.

Programs

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

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000040(k).

A302239 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 6, 16, 40, 96, 226, 512, 1140, 2488, 5336, 11270, 23494, 48356, 98438, 198338, 395846, 783136, 1536800, 2992818, 5786952, 11114950, 21213906, 40247696, 75928804, 142475644, 265985628, 494155176, 913802164, 1682338192, 3084101744, 5630853218, 10240484332, 18553818210
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Comments

Convolution of the sequences A001970 and A261049.

Links

Crossrefs

Cf. A000041, A001970, A015128, A156616, A206622, A206623, A206624, A260916, A261049, A261386, A261452, A261519, A261520, A301554, A301555, A302237, A302238.

Programs

  • Mathematica
    nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000041(k).

A321389 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(k^k).

Original entry on oeis.org

1, 2, 10, 72, 670, 7896, 113532, 1938948, 38463150, 869969602, 22098936536, 622728174288, 19271479902324, 649553475002720, 23680210649058960, 928276725059295192, 38931910620358040382, 1739307894106738293052, 82457731356894087128054, 4134332188240252347401752, 218571692793801915329820184
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 08 2018

Keywords

Comments

Convolution of A023880 and A261053.

Links

Crossrefs

Cf. A023880, A156616, A206622, A206623, A206624, A261053.

Programs

  • Maple
    a:=series(mul(((1+x^k)/(1-x^k))^(k^k),k=1..100),x=0,21): seq(coeff(a,x,n),n=0..20); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(k^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[((-1)^(k/d + 1) + 1) d^(d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 20}]
  • PARI
    seq(n)={Vec(exp(sum(k=1, n, sumdiv(k,d, ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Nov 09 2018

Formula

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} ((-1)^(k/d+1) + 1)*d^(d+1) ) * x^k/k).
a(n) ~ 2 * n^n * (1 + 2*exp(-1)/n + (exp(-1) + 10*exp(-2))/n^2). - Vaclav Kotesovec, Nov 09 2018
Previous Showing 11-14 of 14 results.

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