A306485 Expansion of Product_{k>=1} 1/(1 - Catalan(k)*x^k), where Catalan = A000108.
1, 1, 3, 8, 26, 78, 271, 874, 3096, 10537, 37884, 132282, 484369, 1723568, 6362479, 23042165, 85706354, 313629597, 1175860079, 4340963778, 16355209663, 60882536222, 230370880224, 862533878347, 3278709952956, 12337333292318, 47042968508785, 177882993705004, 680221802560835, 2581438941995517
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Programs
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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}]
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