OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: exp(Sum_{k>=1} Sum_{j>=1} Catalan(j)^k*x^(j*k)/k).
a(n) ~ c * 4^n / (sqrt(Pi)*n^(3/2)), where c = Product_{k>=1} 1/(1 - Catalan(k) / 4^k) = 2.868839868502632... - Vaclav Kotesovec, Feb 23 2019
MAPLE
C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
b(n, i-1)+C(i)*b(n-i, min(n-i, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Aug 23 2019
MATHEMATICA
nmax = 29; CoefficientList[Series[Product[1/(1 - CatalanNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[Sum[CatalanNumber[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d CatalanNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 18 2019
STATUS
approved