A092956 - OEIS

%I #26 Aug 12 2022 00:34:15

%S 1,8,90,1344,25200,570240,15135120,461260800,15878903040,609493248000,

%T 25812039052800,1195656969830400,60138698780160000,

%U 3264143527636992000,190165504623494400000,11836497605427855360000,783921372659482337280000

%N a(n) = (2*n+2)!/((n+2)*n!).

%H G. C. Greubel, <a href="/A092956/b092956.txt">Table of n, a(n) for n = 0..350</a>

%F a(n) = Sum_{k=1..n+1} Gamma(n+1+k)/Gamma(k). - _Bruno Berselli_, Mar 06 2013

%F Let E(x) = Sum_{n>=0} a(n)*x^(2*n)/n!, then E(x) = 2- E(0,x), where E(k,x) = 1 - x^2*(k+1)/( x^2*(k+1) + (k + 1 -x^2)*(k + 2 -x^2)/E(k+1,x) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 21 2013

%F a(n) = A092582(2n+2, n+1). - _Alois P. Heinz_, Jun 19 2017

%F From _G. C. Greubel_, Aug 11 2022: (Start)

%F G.f.: Hypergeometric2F1([2,2,3/2], [3], 4*x).

%F E.g.f.: 4*x*Hypergeometric2F1([5/2,3], [4], 4*x) + Hypergeometric2F1([3/2,2], [3], 4*x). (End)

%p seq((2*n+2)!/(n+2)/n!,n=0..17); # _Emeric Deutsch_

%p a:=n->sum(mul (j-k+n,j=1..n),k=1..n): seq(a(n),n=1..17); # _Zerinvary Lajos_, Jun 04 2007

%t Table[(2n+2)!/((n+2) n!), {n, 0, 16}] (* _Bruno Berselli_, Mar 06 2013 *)

%o (Maxima) A092956(n):=(2*n+2)!/((n+2)*n!)$ makelist(A092956(n),n,0,30); /* _Martin Ettl_, Nov 05 2012 */

%o (Magma) [Factorial(n+1)*Binomial(2*n+2, n): n in [0..20]]; // _G. C. Greubel_, Aug 11 2022

%o (SageMath) [factorial(n+1)*binomial(2*n+2,n) for n in (0..20)] # _G. C. Greubel_, Aug 11 2022

%Y Row sums of A105725.

%Y Cf. A092582.

%K easy,nonn

%O 0,2

%A _Amarnath Murthy_, Mar 25 2004

%E More terms from _Emeric Deutsch_, Apr 18 2005

%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, May 27 2007

%E More terms from _Zerinvary Lajos_, Jun 04 2007