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

%I #12 Sep 08 2022 08:45:09

%S 1,5,24,134,895,7041,63840,654900,7491573,94470925,1301130776,

%T 19423173210,312256205651,5376809244457,98700795776640,

%U 1923638785344456,39661911384761865,862362968121278037,19717031047061570776,472849461034147171790,11866892471399392308231

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

%H G. C. Greubel, <a href="/A080996/b080996.txt">Table of n, a(n) for n = 2..440</a>

%F From _Vaclav Kotesovec_, Jul 05 2018: (Start)

%F 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).

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

%F a(n) = Sum_{k=0..n-2} (n!*(n-1)!)/((k+2)!*(n-k-1)!*(n-k-2)!). - _G. C. Greubel_, Jul 15 2019

%t Table[Binomial[n, 2]*HypergeometricPFQ[{1, -n + 1, -n + 2}, {3}, 1], {n, 2, 30}] (* _Vaclav Kotesovec_, Jul 05 2018 *)

%o (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

%o (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

%o (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

%o (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

%K nonn

%O 2,2

%A _Karol A. Penson_, Feb 28 2003

%E Terms a(19) onward added by _G. C. Greubel_, Jul 15 2019