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Expansion of Product_{k>=1} (1 + x^k)^(binomial(2*k,k)/(k+1)).
3

%I #6 Mar 21 2021 09:11:51

%S 1,1,2,7,20,67,222,758,2617,9189,32554,116494,420046,1525221,5571065,

%T 20457808,75476447,279636977,1039965746,3880891892,14527657602,

%U 54537434161,205270200229,774460385687,2928429307876,11095878177649,42122749335654,160192845018335,610224764470011

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

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>

%F G.f.: Product_{k>=1} (1 + x^k)^A000108(k).

%F 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 nmax = 28; CoefficientList[Series[Product[(1 + x^k)^CatalanNumber[k], {k, 1, nmax}], {x, 0, nmax}], x]

%t 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 Cf. A000108, A052805, A052854, A088327, A292668.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 18 2018