A336292 -id:A336292 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jul 16 2020

nonn

Cf. A002104, A002720, A006040, A336292.

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

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

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

A346411 a(n) = (n!)^2 * Sum_{k=0..n-1} (-1)^k / ((n-k) * k!)^2.

Original entry on oeis.org

0, 1, -3, 4, -8, 1, 353, 27224, 1871840, 147012849, 13684928021, 1514370713340, 197964773810648, 30300949591876913, 5380510834911767033, 1098630080602791984784, 255851291397441057781120, 67450889282916741495608737, 19994198644782014829579657837, 6623096362909598587714211804212

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

sign

Cf. A002741, A073701, A336292, A340789, A346410.

Table[(n!)^2 Sum[(-1)^k/((n - k) k!)^2, {k, 0, n - 1}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[PolyLog[2, x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2

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

cp's OEIS Frontend

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

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