A142704 A generalized factorial level recursion of a Padovan type: a(n) = b(n)*(a(n-2) + a(n-3)) with b(n) = b(n-1) + k and k=2.
0, 1, 1, 6, 16, 70, 264, 1204, 5344, 26424, 130960, 698896, 3777216, 21576256, 125331136, 760604160, 4701036544, 30121800064, 196619065344, 1323267791104, 9069634616320, 63835247970816, 457287705926656
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
-
Maple
A142704 := proc(n) option remember: if n=0 then 0 elif n=1 then 1 elif n =2 then 1 elif n>=3 then 2*n*(procname(n-2) + procname(n-3)) fi: end: seq(A142704(n), n=0..22); # Johannes W. Meijer, Jul 27 2011
-
Mathematica
Clear[a, b, n, k]; k = 2; b[0] = 0; b[n_] := b[n] = b[n - 1] + k; a[0] = 0; a[1] = 1; a[2] = 2; a[n_] := a[n] = b[n]*(a[n - 2] + a[n - 3]); Table[a[n], {n, 0, 22}] FullSimplify[CoefficientList[Series[Pi/(4*Sqrt[2])*E^(x^2/2)*x *Sqrt[1+x] *(BesselI[-1/4,1/2*(1+x)^2]*(2*BesselI[-3/4,1/2] - BesselI[1/4,1/2]) + BesselI[1/4,1/2*(1+x)^2]*(BesselI[-1/4,1/2] - 2*BesselI[3/4,1/2])), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 28 2012 *)
Formula
a(n) = b(n)*(a(n-2) + a(n-3)) with b(n) = b(n-1) + k and k = 2.
a(n) = 2*n*(a(n-2) + a(n-3)) with a(0) = 0, a(1) = a(2) = 1. - Johannes W. Meijer, Jul 27 2011
From Vaclav Kotesovec, Dec 28 2012: (Start)
E.g.f.: (Pi/(4*sqrt(2)))*exp(x^2/2)*x*sqrt(1+x)*(BesselI(-1/4,1/2*(1+x)^2)*(2*BesselI(-3/4,1/2)-BesselI(1/4,1/2))+BesselI(1/4,1/2*(1+x)^2)*(BesselI(-1/4,1/2)-2*BesselI(3/4,1/2))).
a(n) ~ (sqrt(Pi)/8) * (2*BesselI(-3/4,1/2) - 2*BesselI(3/4,1/2) + BesselI(-1/4,1/2) - BesselI(1/4,1/2)) * 2^(n/2-1/4)*exp(sqrt(n)/sqrt(2)-n/2+3/8)*n^(n/2+1/4) * (1-47/(48*sqrt(2*n))). (End)
Extensions
Edited and information added by Johannes W. Meijer, Jul 27 2011