A281591 -id:A281591 - cp's OEIS frontend

A284896 Expansion of Product_{k>=1} 1/(1+x^k)^(k^2) in powers of x.

1, -1, -3, -6, 0, 11, 42, 63, 73, -45, -267, -720, -1095, -1239, -66, 2794, 8757, 16017, 22885, 19634, -2359, -61979, -161867, -302190, -421971, -432051, -126712, 690578, 2278273, 4584989, 7269985, 8965464, 7515373, -845659, -19930400, -53474765, -100195759

Offset: 0

Seiichi Manyama, Apr 05 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n^2, g(n) = -1. - Seiichi Manyama, Nov 15 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A027998, A281590, A281591.

Product_{k>=1} 1/(1+x^k)^(k^m): A081362 (m=0), A255528 (m=1), this sequence (m=2), A284897 (m=3), A284898 (m=4), A284899 (m=5).

CoefficientList[Series[Product[1/(1 + x^k)^(k^2) , {k, 40}], {x, 0, 40}], x] (* Indranil Ghosh, Apr 05 2017 *)

x= 'x + O('x^40); Vec(prod(k=1, 40, 1/(1 + x^k)^(k^2))) \\ Indranil Ghosh, Apr 05 2017

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078307(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

G.f.: exp(Sum_{k>=1} (-1)^k*x^k*(1 + x^k)/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 30 2018

A281590 Indices k such that A284896(k-1) and A284896(k) have a different sign.

Original entry on oeis.org

1, 4, 9, 15, 20, 27, 33, 41, 48, 56, 64, 72, 80, 89, 98, 107, 116, 126, 136, 146, 156, 166, 176, 187, 198, 208, 219, 231, 242, 253, 265, 276, 288, 300, 312, 324, 337, 349, 362, 374, 387, 400, 413, 426, 439, 452, 465, 479, 492, 506, 519, 533, 547, 561, 575

Offset: 1

Vaclav Kotesovec, Apr 14 2017

nonn

			A284896(8) = 73, A284896(9) = -45, sign is changed, so 9 is in the sequence.
A284896(14) = -66, A284896(15) = 2794, sign is changed, so 15 is in the sequence.

Vaclav Kotesovec, Table of n, a(n) for n = 1..2612

Cf. A284896, A281591.

nmax = 1000; A284896 = Rest[CoefficientList[Series[Product[1/(1 + x^k)^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]]; csign = {1}; Do[If[(A284896[[n]] < 0 && A284896[[n+1]] >= 0) || (A284896[[n]] >= 0 && A284896[[n+1]] < 0), csign = Flatten[{csign, n + 1}]], {n, 1, Length[A284896] - 1}]; csign

cp's OEIS Frontend

A284896 Expansion of Product_{k>=1} 1/(1+x^k)^(k^2) in powers of x.

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