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A306485 Expansion of Product_{k>=1} 1/(1 - Catalan(k)*x^k), where Catalan = A000108.

%I A306485 #10 Aug 23 2019 16:21:38
%S A306485 1,1,3,8,26,78,271,874,3096,10537,37884,132282,484369,1723568,6362479,
%T A306485 23042165,85706354,313629597,1175860079,4340963778,16355209663,
%U A306485 60882536222,230370880224,862533878347,3278709952956,12337333292318,47042968508785,177882993705004,680221802560835,2581438941995517
%N A306485 Expansion of Product_{k>=1} 1/(1 - Catalan(k)*x^k), where Catalan = A000108.
%H A306485 Vaclav Kotesovec, <a href="/A306485/b306485.txt">Table of n, a(n) for n = 0..1000</a>
%F A306485 G.f.: exp(Sum_{k>=1} Sum_{j>=1} Catalan(j)^k*x^(j*k)/k).
%F A306485 a(n) ~ c * 4^n / (sqrt(Pi)*n^(3/2)), where c = Product_{k>=1} 1/(1 - Catalan(k) / 4^k) = 2.868839868502632... - _Vaclav Kotesovec_, Feb 23 2019
%p A306485 C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
%p A306485 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A306485       b(n, i-1)+C(i)*b(n-i, min(n-i, i)))
%p A306485     end:
%p A306485 a:= n-> b(n$2):
%p A306485 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 23 2019
%t A306485 nmax = 29; CoefficientList[Series[Product[1/(1 - CatalanNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A306485 nmax = 29; CoefficientList[Series[Exp[Sum[Sum[CatalanNumber[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
%t A306485 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d CatalanNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]
%Y A306485 Cf. A000108, A088327, A179381.
%K A306485 nonn
%O A306485 0,3
%A A306485 _Ilya Gutkovskiy_, Feb 18 2019