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A309682 G.f.: C(x)*C(2*x^2)*C(3*x^3)*..., where C(x) is the g.f. for A000108.

%I A309682 #14 Aug 24 2019 16:01:10
%S A309682 1,1,4,10,33,81,282,762,2599,7979,27343,89371,315256,1078498,3857048,
%T A309682 13651786,49475282,178736186,655247192,2401663838,8883371016,
%U A309682 32906649488,122619768860,457836275272,1716620421629,6449729802639,24308647131627,91800114425437
%N A309682 G.f.: C(x)*C(2*x^2)*C(3*x^3)*..., where C(x) is the g.f. for A000108.
%H A309682 Alois P. Heinz, <a href="/A309682/b309682.txt">Table of n, a(n) for n = 0..1669</a>
%F A309682 a(n) ~ c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (2^k*(2^(k-1) - sqrt(4^(k-1) - k))/k) = 0.711438694828613555153724789...
%p A309682 C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
%p A309682 b:= proc(n, i) option remember; `if`(n=0 or i=1,
%p A309682       C(n), add(C(j)*i^j*b(n-i*j, i-1), j=0..n/i))
%p A309682     end:
%p A309682 a:= n-> b(n$2):
%p A309682 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 23 2019
%t A309682 nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]
%t A309682 nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*k*x^k])/(2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A309682 Cf. A025174, A110143, A322204, A326985.
%K A309682 nonn
%O A309682 0,3
%A A309682 _Vaclav Kotesovec_, Aug 12 2019