A065948 Bessel polynomial {y_n}''(-3).

0, 0, 6, -240, 9540, -415590, 20134590, -1082674404, 64221641820, -4173853100670, 295282282905720, -22605059036265420, 1862664627479732076, -164425432052147568120, 15483794266369962976170, -1549617160894627918342620, 164264715996348003982855020

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..340
• 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]*(-6)^(n - 2)* Hypergeometric1F1[2 - n, -2*n, -2/3], {n, 2, 50}]] (* G. C. Greubel, Aug 15 2017 *)

PARI

for(n=0,50, print1(sum(k=0,n-2, ((n+k+2)!/(4*k!*(n-k-2)!))*(-3/2)^k ), ", ")) \\ G. C. Greubel, Aug 15 2017

Formula

From G. C. Greubel, Aug 15 2017: (Start)

a(n) = 4*n*(n - 1)*(1/2){n}*(-6)^(n - 2)* hypergeometric1f1(2 - n; -2*n; -2/3), where (a){n} is the Pochhammer symbol.

E.g.f.: (-1/81)*(1 + 6*x)^(-5/2)*((-99*x^2 - 54*x - 4)*sqrt(1 + 6*x) + (-54*x^3 + 66*x + 4))*exp(-(1 - sqrt(1 + 6*x))/3). (End)

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