A065946 Bessel polynomial {y_n}''(-2).

0, 0, 6, -150, 3870, -110670, 3538500, -125941284, 4953759300, -213744815460, 10047637214010, -511403305348650, 28029852267603186, -1646397200571955650, 103190849406195456360, -6875135229835376875560, 485256294032090950981800

Offset: 0

Author

N. J. A. Sloane, Dec 08 2001

Keywords

• sign

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)!))*(-1)^k), ", ")) \\ 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), where (a){n} is the Pochhammer symbol.

E.g.f.: (1/16)*(1 + 4*x)^(-5/2)*((24*x^2 + 20*x + 2)*sqrt(1 + 4*x) + (16*x^3 - 12*x^2 - 24*x - 2))*exp((sqrt(1 + 4*x) -1)/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