A256506 a(n) = (2*n+3)*a(n-1) + a(n-2), a(0)=0, a(1)=1.

0, 1, 7, 64, 711, 9307, 140316, 2394679, 45639217, 960818236, 22144458645, 554572284361, 14995596136392, 435426860239729, 13513228263567991, 446371959557983432, 15636531812792988111, 578998049032898543539, 22596560444095836186132, 927037976256962182174951

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

Formula

a(n) = i*(BesselK[5/2,1]*BesselI[n+5/2,-1] - BesselI[5/2,-1]*BesselK[n+5/2,1]) for n >= 0.

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.

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))).

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

a(n) ~ (exp(2)-7) * 2^(n+3/2) * n^(n+2) / exp(n+1). - Vaclav Kotesovec, Jul 22 2015

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-2]+(2n+3)a[n-1]}, a, {n, 20}]
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 *)

Prog

(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

Crossrefs

Sequence in context: A213515 A349333 A293470 * A008787 A261500 A173516

Adjacent sequences: A256503 A256504 A256505 * A256507 A256508 A256509

Keyword

easy,nonn

Author

G. C. Greubel, Apr 22 2015

Extensions

More terms from Vaclav Kotesovec, Jul 22 2015

Status

approved