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

1, 1, 2, 7, 20, 67, 222, 758, 2617, 9189, 32554, 116494, 420046, 1525221, 5571065, 20457808, 75476447, 279636977, 1039965746, 3880891892, 14527657602, 54537434161, 205270200229, 774460385687, 2928429307876, 11095878177649, 42122749335654, 160192845018335, 610224764470011

Offset: 0

Author

Ilya Gutkovskiy, May 18 2018

Keywords

• nonn

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Catalan Number

Crossrefs

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

Programs

Mathematica

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

Formula

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

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