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A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.

%I A318264 #14 Aug 23 2019 16:19:40
%S A318264 1,1,2,7,19,66,212,743,2487,9012,31177,113775,404584,1490726,5376676,
%T A318264 20028981,73068861,273659672,1009921813,3801386137,14125670266,
%U A318264 53477758556,199950414035,759566205693,2857261603610,10889590477287,41136917417501,157329747348492
%N A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.
%H A318264 Vaclav Kotesovec, <a href="/A318264/b318264.txt">Table of n, a(n) for n = 0..1650</a>
%F A318264 a(n) ~ c * A000108(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + C(k)/4^k) = 2.608465265690846547082817204714986077801494... - _Vaclav Kotesovec_, Aug 24 2018
%p A318264 C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
%p A318264 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A318264      `if`(n=0, 1, b(n, i-1)+C(i)*b(n-i, min(n-i, i-1))))
%p A318264     end:
%p A318264 a:= n-> b(n$2):
%p A318264 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 23 2019
%t A318264 nmax = 40; CoefficientList[Series[Product[1+CatalanNumber[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A318264 nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += CatalanNumber[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly
%Y A318264 Cf. A000108, A022629, A179381, A318248, A318263.
%K A318264 nonn
%O A318264 0,3
%A A318264 _Vaclav Kotesovec_, Aug 22 2018