A080996 Special values of the hypergeometric function 3F1: a(n) = binomial(n,2) * hypergeom([1,-n+1,-n+2],[3],1).

1, 5, 24, 134, 895, 7041, 63840, 654900, 7491573, 94470925, 1301130776, 19423173210, 312256205651, 5376809244457, 98700795776640, 1923638785344456, 39661911384761865, 862362968121278037, 19717031047061570776, 472849461034147171790, 11866892471399392308231

Offset

2,2

Links

G. C. Greubel, Table of n, a(n) for n = 2..440

Formula

From Vaclav Kotesovec, Jul 05 2018: (Start)

Recurrence: (n-2)*(n+1)*(n^2 - 7*n + 9)*a(n) = 2*(n-1)*(n^4 - 8*n^3 + 16*n^2 - 5*n - 9)*a(n-1) - (n-2)*(n-1)*(n^4 - 8*n^3 + 21*n^2 - 30*n + 15)*a(n-2) + (n-3)*(n-2)^2*(n-1)*(n^2 - 5*n + 3)*a(n-3).

a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n - 5/4) / sqrt(2). (End)

a(n) = Sum_{k=0..n-2} (n!*(n-1)!)/((k+2)!*(n-k-1)!*(n-k-2)!). - G. C. Greubel, Jul 15 2019

Mathematica

Table[Binomial[n, 2]*HypergeometricPFQ[{1, -n + 1, -n + 2}, {3}, 1], {n, 2, 30}] (* Vaclav Kotesovec, Jul 05 2018 *)

Prog

(PARI) vector(30, n, n++; sum(k=0, n-2, (n!*(n-1)!)/((k+2)!*(n-k-1)!*(n-k-2)!)) ) \\ G. C. Greubel, Jul 15 2019
(Magma) F:=Factorial; [(&+[(F(n)*F(n-1))/(F(k+2)*F(n-k-1)*F(n-k-2)): k in [0..n-2]]): n in [2..30]]; // G. C. Greubel, Jul 15 2019
(Sage) f=factorial; [sum((f(n)*f(n-1))/(f(k+2)*f(n-k-1)*f(n-k-2)) for k in (0..n-2)) for n in (2..30)] # G. C. Greubel, Jul 15 2019
(GAP) F:=Factorial;; List([2..30], n-> Sum([0..n-2], k-> (F(n)*F(n-1))/( F(k+2)*F(n-k-1)*F(n-k-2)) )); # G. C. Greubel, Jul 15 2019

Crossrefs

Sequence in context: A002709 A193668 A009411 * A020055 A256325 A286743

Adjacent sequences: A080993 A080994 A080995 * A080997 A080998 A080999

Keyword

nonn

Author

Karol A. Penson, Feb 28 2003

Extensions

Terms a(19) onward added by G. C. Greubel, Jul 15 2019

Status

approved