A346315 Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} 1 / (1 + (-x)^n / (n!)^2).

1, 1, 3, 28, 483, 11976, 423660, 20801775, 1337182819, 108259612048, 10814058518328, 1308659192928495, 188498906179378476, 31855351764833425895, 6243218508505581436249, 1404734813476218805338303, 359618310105650201828166499, 103929494668760259335327432160

Offset: 0

Author

Ilya Gutkovskiy, Jul 13 2021

Keywords

• nonn

Crossrefs

Cf. A076901, A183229, A183240, A292308, A346314.

Programs

Mathematica

nmax = 17; CoefficientList[Series[Product[1/(1 + (-x)^k/(k!)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[(-1)^k (Binomial[n, k] k!)^2 k Sum[(-1)^d/(d ((k/d)!)^(2 d)), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

Formula

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