A344108 Expansion of Product_{k>=1} 1 / (1 - x^k)^binomial(2*k,k).

1, 2, 9, 36, 154, 644, 2744, 11608, 49267, 208610, 882963, 3731640, 15754327, 66426946, 279766063, 1176920484, 4945739292, 20761707824, 87069433162, 364802647912, 1527072152856, 6386873581244, 26690795165394, 111453873957936, 465055114353616, 1939114409985956

Offset: 0

Author

Ilya Gutkovskiy, May 09 2021

Keywords

• nonn

Comments

Euler transform of A000984.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000984, A005430, A088327, A194353, A344130.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
      binomial(2*d, d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 13 2023

Mathematica

nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^Binomial[2 k, k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d Binomial[2 d, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Exp[Sum[(1/Sqrt[1 - 4*x^j] - 1)/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 10 2021 *)

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} A005430(d) ) * a(n-k).

a(n) ~ 2^(2*n - 1/3) * exp(3*n^(1/3)/2^(2/3) - 1 + c) / (sqrt(3*Pi) * n^(5/6)), where c = Sum_{k>=2} (1/sqrt(1 - 4^(1-k)) - 1)/k = 0.0907540019413286886324751305813463657179452545... - Vaclav Kotesovec, May 10 2021