This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A142704 #17 Mar 03 2019 21:43:50
%S A142704 0,1,1,6,16,70,264,1204,5344,26424,130960,698896,3777216,21576256,
%T A142704 125331136,760604160,4701036544,30121800064,196619065344,
%U A142704 1323267791104,9069634616320,63835247970816,457287705926656
%N 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.
%H A142704 Vincenzo Librandi, <a href="/A142704/b142704.txt">Table of n, a(n) for n = 0..200</a>
%F A142704 a(n) = b(n)*(a(n-2) + a(n-3)) with b(n) = b(n-1) + k and k = 2.
%F A142704 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
%F A142704 From _Vaclav Kotesovec_, Dec 28 2012: (Start)
%F A142704 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))).
%F A142704 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)
%p A142704 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
%t A142704 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}]
%t A142704 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 *)
%Y A142704 Cf. A171386 (k=0), A108189 (k=1), A002467 (Game of Mousetrap), A000931 (Padovan).
%K A142704 nonn,easy
%O A142704 0,4
%A A142704 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2008
%E A142704 Edited and information added by _Johannes W. Meijer_, Jul 27 2011