|
%I #32 Apr 29 2023 00:05:54
%S 0,1,7,64,711,9307,140316,2394679,45639217,960818236,22144458645,
%T 554572284361,14995596136392,435426860239729,13513228263567991,
%U 446371959557983432,15636531812792988111,578998049032898543539,22596560444095836186132,927037976256962182174951
%N a(n) = (2*n+3)*a(n-1) + a(n-2), a(0)=0, a(1)=1.
%H G. C. Greubel, <a href="/A256506/b256506.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = i*(BesselK[5/2,1]*BesselI[n+5/2,-1] - BesselI[5/2,-1]*BesselK[n+5/2,1]) for n >= 0.
%F a(n) = (2/Pi)*(i_{2}^{(1)}(-1)*k_{n+2}(1) - k_{2}(1)*i_{n+2}^{(1)}(-1)) where i_{n}^{(1)}(x) and k_{n}(x) are the modified spherical Bessel functions, n >= 0.
%F E.g.f.: Sum_{n>=0} a(n-2) t^n/n! = (1/(2*e*sqrt(1-2t)))*((e^2 - 7)*cosh(sqrt(1-2t)) - (e^2 + 7)*sinh(sqrt(1-2t))).
%F a(n) = (exp(2)-7)*BesselK(5/2 + n, 1)/(exp(1)*sqrt(2*Pi)) - 7*(-1)^n*sqrt(Pi/2) * BesselI(5/2 + n, 1)/exp(1). - _Vaclav Kotesovec_, Jul 22 2015
%F a(n) ~ (exp(2)-7) * 2^(n+3/2) * n^(n+2) / exp(n+1). - _Vaclav Kotesovec_, Jul 22 2015
%t RecurrenceTable[{a[0]==0,a[1]==1, a[n]==a[n-2]+(2n+3)a[n-1]}, a, {n, 20}]
%t nxt[{n_,a_,b_}]:={n+1,b,(2n+5)*b+a}; NestList[nxt,{1,0,1},20][[All,2]] (* _Harvey P. Dale_, Jan 07 2021 *)
%o (Magma) [n le 2 select n-1 else (2*n+1)*Self(n-1)+Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, May 02 2016
%K easy,nonn
%O 0,3
%A _G. C. Greubel_, Apr 22 2015
%E More terms from _Vaclav Kotesovec_, Jul 22 2015
|
