A384351 Expansion of Product_{k>=1} 1/(1 - k*(k+1)/2 * x)^((1/2)^(k+2)).

1, 1, 7, 143, 6140, 455828, 51947988, 8414718996, 1836791273514, 519582028795210, 184852108308617398, 80776494267416227078, 42529172631705836804876, 26553065315757661351020284, 19397441882229095276127402500, 16390942374821715002096327774628

Offset: 0

Author

Seiichi Manyama, May 27 2025

Keywords

• nonn

Crossrefs

Cf. A084784, A384352, A384353.

Cf. A055203, A262809.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[Sum[Sum[Sum[(-1)^j*Binomial[i, j]*((i - j)*(i - j - 1)/2)^k, {j, 0, i}], {i, 0, 2 k}]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 29 2025 *)

PARI

a262809(n, k) = sum(i=0, k*n, sum(j=0, i, (-1)^j*binomial(i, j)*binomial(i-j, n)^k));
my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, a262809(2, k)*x^k/k)))

Formula

G.f.: exp(Sum_{k>=1} A055203(k) * x^k/k).

a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(2*n - 1/2) / (exp(2*n) * log(2)^(2*n+1)). - Vaclav Kotesovec, May 29 2025