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A304787 Expansion of Product_{k>=1} (1 + x^k)^(binomial(2*k,k)/(k+1)).

%I A304787 #10 Feb 16 2025 08:33:54
%S A304787 1,1,2,7,20,67,222,758,2617,9189,32554,116494,420046,1525221,5571065,
%T A304787 20457808,75476447,279636977,1039965746,3880891892,14527657602,
%U A304787 54537434161,205270200229,774460385687,2928429307876,11095878177649,42122749335654,160192845018335,610224764470011
%N A304787 Expansion of Product_{k>=1} (1 + x^k)^(binomial(2*k,k)/(k+1)).
%H A304787 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A304787 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>
%F A304787 G.f.: Product_{k>=1} (1 + x^k)^A000108(k).
%F A304787 a(n) ~ c * 4^n / n^(3/2), where c = exp( Sum_{k>=1} (-1)^k * (2 - 4^k + 4^k*sqrt(1 - 4^(1-k)))/(2*k) ) / sqrt(Pi) = 1.4863036894111457491052224706533674748514957... - _Vaclav Kotesovec_, Mar 21 2021
%t A304787 nmax = 28; CoefficientList[Series[Product[(1 + x^k)^CatalanNumber[k], {k, 1, nmax}], {x, 0, nmax}], x]
%t A304787 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d CatalanNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]
%Y A304787 Cf. A000108, A052805, A052854, A088327, A292668.
%K A304787 nonn
%O A304787 0,3
%A A304787 _Ilya Gutkovskiy_, May 18 2018