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A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).

%I A266821 #17 Aug 28 2019 13:51:27
%S A266821 1,4,8,24,44,88,176,312,544,924,1584,2552,4136,6488,10128,15632,23748,
%T A266821 35640,53080,78136,114024,165552,237744,339544,481248,678236,949008,
%U A266821 1321840,1830376,2521688,3456672,4717208,6406680,8666448,11672464,15660528,20934868
%N A266821 Expansion of Product_{k>=1} (1 + 3*x^k) / (1 - x^k).
%C A266821 Convolution of A000041 and A032308.
%C A266821 In general, for m > 0, if g.f. = Product_{k>=1} ((1 + m*x^k) / (1 - x^k)) then a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(m+1)*n), where c = 2*Pi^2/3 + log(m)^2 + 2*polylog(2, -1/m).
%H A266821 Alois P. Heinz, <a href="/A266821/b266821.txt">Table of n, a(n) for n = 0..10000</a> (first 5001 terms from Vaclav Kotesovec)
%F A266821 a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (8*Pi*n), where c = 2*Pi^2/3 + log(3)^2 + 2*polylog(2, -1/3) = 7.16861897522987077909937377164783326088308015803... .
%p A266821 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A266821       (t-> b(t, min(t, i-1)))(n-i*j), j=1..n/i)*4 +b(n, i-1)))
%p A266821     end:
%p A266821 a:= n-> b(n$2):
%p A266821 seq(a(n), n=0..44);  # _Alois P. Heinz_, Aug 28 2019
%t A266821 nmax = 40; CoefficientList[Series[Product[(1+3*x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%o A266821 (PARI) { my(n=40); Vec(prod(k=1, n, 4/(1-x^k) - 3 + O(x*x^n))) } \\ _Andrew Howroyd_, Dec 22 2017
%Y A266821 Cf. A000041, A015128, A032308, A264686.
%Y A266821 Column k=4 of A321884.
%K A266821 nonn
%O A266821 0,2
%A A266821 _Vaclav Kotesovec_, Jan 04 2016