A336292 a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(n-k) / (k * ((n-k)!)^2).

0, 1, -2, 3, 8, 305, 10734, 502747, 30344992, 2307890097, 216571514030, 24619605092291, 3337294343698248, 532148381719443073, 98646472269855762238, 21041945289232131607995, 5118447176652195630775424, 1408601897794844346184122017, 435481794298015565250651718302

Offset: 0

Author

Ilya Gutkovskiy, Jul 16 2020

Keywords

• sign

Crossrefs

Cf. A002741, A009940, A336291.

Programs

Mathematica

Table[(n!)^2 Sum[(-1)^(n - k)/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[1 - x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

PARI

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

Formula

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