A051893 a(n) = Sum_{i=1..n-1} i^2*a(i), a(1) = 1.

1, 1, 5, 50, 850, 22100, 817700, 40885000, 2657525000, 217917050000, 22009622050000, 2685173890100000, 389350214064500000, 66189536390965000000, 13039338669020105000000, 2946890539198543730000000, 757350868574025738610000000, 219631751886467464196900000000

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 17 1999

Keywords

• nice
• nonn

Crossrefs

Cf. A001710, A101686, A256019, A256020.

Programs

Maple

a := n -> `if`(n=1,1,(sinh(Pi)*GAMMA(n-I)*GAMMA(n+I))/(2*Pi)):
seq(simplify(a(n)), n=1..18); # Peter Luschny, Oct 19 2016

Mathematica

a[n_] := Pochhammer[2-I, n-2]*Pochhammer[2+I, n-2]; a[1] = 1; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 21 2012, after Vladeta Jovovic *)
Join[{1},FoldList[Times,1,Range[2,20]^2+1]] (* Harvey P. Dale, Jul 04 2013 *)
Clear[a]; a[1]=1; a[n_]:=a[n]=Sum[i^2*a[i],{i,1,n-1}]; Table[a[n],{n,1,20}] (* Vaclav Kotesovec, Mar 13 2015 *)

Formula

a(n) = Product_{i=2..n-1} (i^2+1), for n>2. - Vladeta Jovovic, Nov 26 2002

From Vaclav Kotesovec, Mar 13 2015: (Start)

For n > 1, a(n) = A101686(n-1)/2.

a(n) ~ (n-1)!^2 * sinh(Pi)/(2*Pi).

(End)

a(n) = (A003703(n)^2 + A009454(n)^2 + A000007(n-1))/2. - Vladimir Reshetnikov, Oct 15 2016

a(n) = sinh(Pi)*Gamma(n-I)*Gamma(n+I)/(2*Pi) for n>1. - Peter Luschny, Oct 19 2016

Extensions

More terms from Harvey P. Dale, Jul 04 2013