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A344108 Expansion of Product_{k>=1} 1 / (1 - x^k)^binomial(2*k,k).

%I A344108 #13 Feb 13 2023 17:59:08
%S A344108 1,2,9,36,154,644,2744,11608,49267,208610,882963,3731640,15754327,
%T A344108 66426946,279766063,1176920484,4945739292,20761707824,87069433162,
%U A344108 364802647912,1527072152856,6386873581244,26690795165394,111453873957936,465055114353616,1939114409985956
%N A344108 Expansion of Product_{k>=1} 1 / (1 - x^k)^binomial(2*k,k).
%C A344108 Euler transform of A000984.
%H A344108 Alois P. Heinz, <a href="/A344108/b344108.txt">Table of n, a(n) for n = 0..1000</a>
%F A344108 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} A005430(d) ) * a(n-k).
%F A344108 a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (1/sqrt(1 - 4^(1-k)) - 1)/k = 0.0907540019413286886324751305813463657179452545... - _Vaclav Kotesovec_, May 10 2021
%p A344108 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
%p A344108       binomial(2*d, d), d=numtheory[divisors](j)), j=1..n)/n)
%p A344108     end:
%p A344108 seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 13 2023
%t A344108 nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]
%t A344108 a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d Binomial[2 d, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
%t A344108 nmax = 25; CoefficientList[Series[Exp[Sum[(1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 10 2021 *)
%Y A344108 Cf. A000984, A005430, A088327, A194353, A344130.
%K A344108 nonn
%O A344108 0,2
%A A344108 _Ilya Gutkovskiy_, May 09 2021