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 1-4 of 4 results.

A294846 Expansion of Product_{k>=1} 1/(1 + x^k)^(k*(k+1)/2).

Original entry on oeis.org

1, -1, -2, -4, 0, 3, 17, 24, 40, 9, -24, -149, -250, -435, -395, -281, 514, 1528, 3542, 5127, 6920, 5416, 1368, -11136, -28533, -57051, -82846, -107315, -95655, -43646, 107826, 345877, 727771, 1150968, 1601729, 1766547, 1495154, 183944, -2339567, -6770991, -12701854
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 09 2017

Keywords

Comments

Convolution inverse of A028377.
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. A000217, A000294, A028377, A255528, A284896, A292386.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Product[1/(1 + x^k)^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 40}]

Formula

G.f.: Product_{k>=1} 1/(1 + x^k)^A000217(k).
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)^(n/d). - Seiichi Manyama, Nov 14 2017

A292387 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)*(k+2)/6).

Original entry on oeis.org

1, -1, -4, -6, -4, 19, 60, 131, 149, -4, -572, -1764, -3485, -4716, -2658, 7606, 32944, 77152, 132586, 161275, 75150, -281687, -1111029, -2560293, -4470415, -5922117, -4603551, 3799070, 25573251, 67259095, 130430051, 201158707, 232853019, 124749892, -295134275, -1260897993, -2995361708, -5515840117
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 15 2017

Keywords

Comments

Convolution inverse of A000335 (Euler transform of the tetrahedral numbers).

Crossrefs

Cf. A000335, A258343, A292386.

Programs

  • Mathematica
    nmax = 37; CoefficientList[Series[Product[(1 - x^k)^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^k)^(k*(k+1)*(k+2)/6).

A305205 a(n) = [x^n] exp(-Sum_{k>=1} x^k/(k*(1 - x^k)^n)).

Original entry on oeis.org

1, -1, -2, -3, -4, 30, 274, 1841, 9358, 32463, -41557, -2265846, -28939286, -272101778, -2038274408, -10494221259, 9056975574, 1244820826687, 22703501504125, 299864024917632, 3221417281127823, 26849622543478562, 110101743392268978, -1810492304600468063
Offset: 0

Views

Author

Ilya Gutkovskiy, May 27 2018

Keywords

Links

Crossrefs

Cf. A073592, A292386, A292387, A293554, A305206.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[-Sum[x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 - x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 23}]

Formula

a(n) = [x^n] Product_{k>=1} (1 - x^k)^binomial(n+k-2,n-1).

A319757 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)*(2*k+1)/6).

Original entry on oeis.org

1, -1, -5, -9, -6, 35, 125, 275, 291, -241, -2111, -5989, -10990, -11660, 6454, 68298, 201859, 400794, 546122, 269907, -1175825, -4890783, -11746437, -20668698, -25146121, -7959643, 63707489, 236244458, 546634684, 956731805, 1220119643, 676723572, -1964409479, -8645307595
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 27 2018

Keywords

Crossrefs

Cf. A000330, A001157, A001158, A001159, A279215, A281156, A283263, A292386, A292387.

Programs

  • Maple
    a:=series(mul((1-x^k)^(k*(k+1)*(2*k+1)/6),k=1..100),x=0,34): seq(coeff(a,x,n),n=0..33); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 33; CoefficientList[Series[Product[(1 - x^k)^(k (2 k + 1) (k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[-Sum[x^k (1 + x^k)/(k (1 - x^k)^4), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d^2 (d + 1) (2 d + 1)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Formula

G.f.: Product_{k>=1} (1 - x^k)^A000330(k).
G.f.: exp(-Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^4)).
G.f.: exp(-Sum_{k>=1} (2*sigma_4(k) + 3*sigma_3(k) + sigma_2(k))*x^k/(6*k)).
Showing 1-4 of 4 results.

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