A319455 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^2.

1, 2, 7, 14, 35, 66, 140, 252, 485, 840, 1512, 2534, 4347, 7084, 11705, 18622, 29862, 46522, 72779, 111310, 170534, 256586, 386101, 572488, 848050, 1240974, 1812979, 2621486, 3782669, 5410360, 7720237, 10932740, 15443120, 21669546, 30327570, 42196022, 58555543, 80832850

Offset: 0

Author

Ilya Gutkovskiy, Sep 19 2018

Keywords

• nonn

Comments

Convolution inverse of A002171.

Self-convolution of A002513.

Convolution of A000041 and A029862.

Euler transform of period 2 sequence [2, 4, ...].

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000041, A001934, A001936, A002171, A002513, A029862.

Programs

Maple

a:=series(mul(1/((1-x^k)*(1-x^(2*k)))^2,k=1..55),x=0,38): seq(coeff(a,x,n),n=0..37); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 37; CoefficientList[Series[Product[1/((1 - x^k)*(1 - x^(2*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^2])^2, {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[2 Sum[(4 DivisorSigma[1, k] - DivisorSigma[1, 2 k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

PARI

seq(n)={Vec(exp(2*sum(k=1, n, (4*sigma(k) - sigma(2*k))*x^k/k) + O(x*x^n)))} \\ Andrew Howroyd, Sep 19 2018

Formula

G.f.: Product_{k>=1} (1 + x^k)^2/(1 - x^(2*k))^4.

G.f.: exp(2*Sum_{k>=1} (4*sigma(k) - sigma(2*k))*x^k/k).

a(n) ~ exp(Pi*sqrt(2*n)) / (2^(13/4)*n^(7/4)). - Vaclav Kotesovec, Sep 14 2021