A216831 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k!.

1, 2, 11, 88, 905, 11246, 162607, 2668436, 48830273, 983353690, 21570885011, 511212091952, 13001401709881, 352856328962918, 10170853073795975, 310093415465876716, 9964607161173899777, 336439048405066012466, 11902368222382731461083, 440122520333417057761160

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

Recurrence: (8*n^2+31*n+21)*a(n+3) - (24*n^3+157*n^2+308*n+162)*a(n+2) + (24*n^4+117*n^3+178*n^2+71*n-18)*a(n+1) - (8*n^2+31*n+30)*(n+1)^3*a(n) = 0.

a(n) ~ n^(n-1/6)/(sqrt(6*Pi)*exp(n+n^(1/3)-3*n^(2/3)-1/3)). - Vaclav Kotesovec, Sep 30 2012

a(n) = hypergeom([-n, -n, -n], [1], -1). - Vladimir Reshetnikov, Sep 28 2016

a(n) = Sum_{k=0..n} binomial(n, k)*|A021009(n, k)|. - Peter Luschny, May 04 2021

Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022

Mathematica

Table[Sum[Binomial[n, k]^3*k!, {k, 0, n}], {n, 0, 25}]
Table[HypergeometricPFQ[{-n, -n, -n}, {1}, -1], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 28 2016 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)^3 * k!); \\ Michel Marcus, May 04 2021

Crossrefs

Cf. A000522, A002720, A000172, A119400, A206178, A021009.

Sequence in context: A107096 A361599 A138739 * A221864 A376871 A197900

Adjacent sequences: A216828 A216829 A216830 * A216832 A216833 A216834

Keyword

nonn

Author

Vaclav Kotesovec, Sep 17 2012

Status

approved