A346313 Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} 1 / (1 + (-x)^n / n^2).

1, 1, 3, 31, 496, 12576, 444736, 22056448, 1406058816, 114618828096, 11405077216704, 1385889578069184, 198961869847145472, 33725910553646229504, 6594186368339077238784, 1487133154121568112705536, 379990326228614750079369216, 110013397755650063836228435968

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

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

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 Sum[(-1)^d/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

Crossrefs

Cf. A249588, A292358, A326864, A346312.

Sequence in context: A212917 A223993 A342206 * A143637 A327227 A360091

Adjacent sequences: A346310 A346311 A346312 * A346314 A346315 A346316

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 13 2021

Status

approved