A065707 Bessel polynomial {y_n}'(-2).

0, 1, -9, 126, -2270, 49995, -1301139, 39066076, -1329148764, 50536328085, -2123542798685, 97722882268506, -4887863677728954, 264025383760041631, -15317578742680490535, 949914821498248213560, -62707584375936061905464, 4390358319593012839913001

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. A001514, A065920, A065921, A065922, A065707, A006199.

Programs

Mathematica

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

PARI

for(n=0,50, print1(sum(k=0,n-1, (n+k+1)!/(2*(n-k-1)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

Formula

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

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

E.g.f.: ((1 + 4*x)^(3/2) - 2*x*(1 + 4*x)^(1/2) - 1)* exp((sqrt(1 + 4*x) -1)/2)/(4*(1 + 4*x)^(3/2)). (End)

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