A065950 Bessel polynomial {y_n}'''(1).

0, 0, 0, 90, 3150, 81900, 1992060, 48771450, 1237774230, 32978969100, 927339227100, 27566149731120, 866148362679600, 28735959507074820, 1005105838958594100, 36999204981675832350, 1430792213377354462530, 58019598569681129648700

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..400
• Index entries for sequences related to Bessel functions or polynomials

Crossrefs

Cf. A001516, A001518.

Programs

Magma

[0,0,0] cat [(&+[Binomial(n-3,k)*Factorial(n+k+3)/(2^(k+3) * Factorial(n-3)): k in [0..n-3]]): n in [3..30]]; // G. C. Greubel, Sep 23 2023

Mathematica

Join[{0,0,0}, Table[6*Binomial[n,3]*Pochhammer[1/2,n]*2^n* Hypergeometric1F1[3-n,-2*n,2], {n,3,50}]] (* G. C. Greubel, Aug 15 2017 *)
CoefficientList[Series[(90*t^3/(1-t)^7)*HypergeometricPFQ[{4, 7/2}, {}, 2*t/(1-t)^2], {t,0,50}], t] (* G. C. Greubel, Aug 16 2017 *)

PARI

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

SageMath

def A065950(n): return sum(binomial(n-3,k)*rising_factorial(n-2,k+6)//2^(k+3) for k in range(n-2))
[A065950(n) for n in range(31)] # G. C. Greubel, Sep 23 2023

Formula

a(n) = 6*binomial(n, 3)*(1/2){n}*2^n*hypergeometric1f1(3-n, -2*n, 2), where (a){n} is the Pochhammer symbol. - G. C. Greubel, Aug 15 2017

G.f.: (90*x^3/(1-x)^7)*hypergeometric2f0(4,7/2; - ; 2*x/(1-x)^2). - G. C. Greubel, Aug 16 2017

a(n) ~ 2^(n + 1/2) * n^(n+3) / exp(n-1). - Vaclav Kotesovec, Jun 09 2019