A346371 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( Sum_{n>=0} x^(2*n+1) / (2*n+1)^2 ).

1, 1, 2, 10, 88, 1496, 34256, 1305872, 57804160, 3960382848, 288097804032, 31177032137472, 3374496463248384, 530644850402565120, 79955455534325999616, 17241179374803330287616, 3448609425518084068048896, 977269122457749276877750272, 250420488297020919542581493760

Offset: 0

Author

Ilya Gutkovskiy, Jul 14 2021

Keywords

• nonn

Crossrefs

Cf. A000246, A074707.

Programs

Mathematica

nmax = 18; CoefficientList[Series[Exp[Sum[x^(2 k + 1)/(2 k + 1)^2, {k, 0, Infinity}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[(Binomial[n, 2 k + 1] (2 k + 1)!)^2 a[n - 2 k - 1]/(2 k + 1), {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=0..floor((n-1)/2)} (binomial(n,2*k+1) * (2*k+1)!)^2 * a(n-2*k-1) / (2*k+1).