A336291 a(n) = (n!)^2 * Sum_{k=1..n} 1 / (k * ((n-k)!)^2).

0, 1, 6, 39, 424, 7905, 227766, 9324511, 512970144, 36452217969, 3247711402870, 354391640998791, 46474986465907176, 7210874466760191409, 1306387103147257800774, 273269900360634449732895, 65363179181419926246184576, 17726298367452515070739268001

Offset: 0

Author

Ilya Gutkovskiy, Jul 16 2020

Keywords

• nonn

Crossrefs

Cf. A002104, A002720, A006040, A336292.

Programs

Mathematica

Table[(n!)^2 Sum[1/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[1 - x] BesselI[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

PARI

a(n) = (n!)^2 * sum(k=1, n, 1 / (k * ((n-k)!)^2)); \\ Michel Marcus, Jul 17 2020

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(1 - x) * BesselI(0,2*sqrt(x)).

a(n) ~ BesselI(0,2) * (n!)^2 / n. - Vaclav Kotesovec, Jul 17 2020

a(n) = Sum_{k=1..n} (k-1)!*k!*binomial(n,k)^2. - Ridouane Oudra, Jun 15 2025