A065945 Bessel polynomial {y_n}''(2).

0, 0, 6, 210, 6390, 201810, 6895140, 257335596, 10489055220, 465303486780, 22363517407770, 1159112646836430, 64499453473280826, 3837361123234687230, 243168894263042103720, 16356164256377393353080, 1164094991704907423494920

Offset: 0

Author

N. J. A. Sloane, Dec 08 2001

Keywords

• nonn

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..360
• Index entries for sequences related to Bessel functions or polynomials

Crossrefs

Cf. A001518, A001516.

Programs

Mathematica

Join[{0, 0}, Table[4*n*(n - 1)*Pochhammer[1/2, n]*4^(n - 2)* Hypergeometric1F1[2 - n, -2*n, 1], {n,2,50}]] (* G. C. Greubel, Aug 14 2017 *)

PARI

for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))), ", ")) \\ G. C. Greubel, Aug 14 2017

Formula

From G. C. Greubel, Aug 14 2017: (Start)

a(n) = 4*n*(n - 1)*(1/2)_{n}*4^(n - 2)*hypergeometric1f1(2-n, -2*n, 1).

E.g.f.: (-1/16)*(1 - 4*x)^(-5/2)*((56*x^2 - 44*x + 6)*sqrt(1 - 4*x) + (16*x^3 - 180*x^2 + 56*x - 6))*exp((1 - sqrt(1 - 4*x))/2). (End)

G.f.: (6*x^2/(1-x)^5)*hypergeometric2f0(3,5/2; - ; 4*x/(1-x)^2). - G. C. Greubel, Aug 16 2017