A336291 -id:A336291 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jul 16 2020

sign

Cf. A002741, A009940, A336291.

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

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

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

A346410 a(n) = (n!)^2 * Sum_{k=0..n-1} 1 / ((n-k) * k!)^2.

Original entry on oeis.org

0, 1, 5, 22, 152, 2001, 45097, 1527506, 71864928, 4466430513, 353828600029, 34770661312190, 4148422395161464, 590479899466175681, 98824492409739430401, 19209838771051338898234, 4291488438323868507946880, 1091819942877526843993466529, 313819508664449992611846900981

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Cf. A002104, A006040, A066998, A193563, A336291, A346411.

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula