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A319455 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(2*k)))^2. 3

%I #14 Jan 04 2024 18:09:48

%S 1,2,7,14,35,66,140,252,485,840,1512,2534,4347,7084,11705,18622,29862,

%T 46522,72779,111310,170534,256586,386101,572488,848050,1240974,

%U 1812979,2621486,3782669,5410360,7720237,10932740,15443120,21669546,30327570,42196022,58555543,80832850

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

%C Convolution inverse of A002171.

%C Self-convolution of A002513.

%C Convolution of A000041 and A029862.

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

%H Andrew Howroyd, <a href="/A319455/b319455.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F a(n) ~ exp(Pi*sqrt(2*n)) / (2^(13/4)*n^(7/4)). - _Vaclav Kotesovec_, Sep 14 2021

%p 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

%t nmax = 37; CoefficientList[Series[Product[1/((1 - x^k)*(1 - x^(2*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 37; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^2])^2, {x, 0, nmax}], x]

%t 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]

%o (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

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Sep 19 2018

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Last modified March 29 08:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)