login
a(n) = Sum_{k=0..n} binomial(n,k)^3 * k!.
5

%I #28 Jun 19 2022 08:29:08

%S 1,2,11,88,905,11246,162607,2668436,48830273,983353690,21570885011,

%T 511212091952,13001401709881,352856328962918,10170853073795975,

%U 310093415465876716,9964607161173899777,336439048405066012466,11902368222382731461083,440122520333417057761160

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

%H Vincenzo Librandi, <a href="/A216831/b216831.txt">Table of n, a(n) for n = 0..200</a>

%F 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.

%F 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

%F a(n) = hypergeom([-n, -n, -n], [1], -1). - _Vladimir Reshetnikov_, Sep 28 2016

%F a(n) = Sum_{k=0..n} binomial(n, k)*|A021009(n, k)|. - _Peter Luschny_, May 04 2021

%F 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

%t Table[Sum[Binomial[n, k]^3*k!, {k, 0, n}], {n, 0, 25}]

%t Table[HypergeometricPFQ[{-n, -n, -n}, {1}, -1], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 28 2016 *)

%o (PARI) a(n) = sum(k=0, n, binomial(n,k)^3 * k!); \\ _Michel Marcus_, May 04 2021

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

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Sep 17 2012