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A264686 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - x^k). 5

%I

%S 1,3,6,15,27,51,93,159,264,432,696,1086,1683,2553,3837,5700,8367,

%T 12147,17505,24972,35361,49728,69402,96243,132657,181782,247692,

%U 335838,453042,608289,813102,1082256,1434519,1894215,2491644,3265869,4265973,5553771,7207167

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

%C Convolution of A000041 and A032302.

%H Vaclav Kotesovec, <a href="/A264686/b264686.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(3)*n), where c = 2*Pi^2/3 + log(2)^2 + 2*polylog(2, -1/2) = 6.163360867463814765670634381079217086937812673723341... . - _Vaclav Kotesovec_, Jan 04 2016

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,

%p `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, 2*b(n-i, i-1))))

%p end:

%p a:= n-> add(b(i$2)*combinat[numbpart](n-i), i=0..n):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Dec 22 2017

%t nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%o (PARI) { my(n=40); Vec(prod(k=1, n, 3/(1-x^k) - 2 + O(x*x^n))) } \\ _Andrew Howroyd_, Dec 22 2017

%Y Cf. A000041, A006951, A015128, A032302, A261584, A264685, A266821.

%Y Column k=3 of A321884.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 21 2015

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)