A065944 Bessel polynomial {y_n}''(-1).

0, 0, 6, -60, 720, -9870, 153510, -2679264, 51934680, -1107917910, 25807660560, -651977992380, 17758547202396, -518856566089680, 16188283372489410, -537210169663283760, 18894951642157260480, -702160022681408982114

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

Crossrefs

Cf. A001518, A001516.

Programs

GAP

f:=Factorial;; Concatenation([0,0], List([2..20], n-> Sum([0..n-2], k-> (-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2)*f(k)) ))); # G. C. Greubel, Jul 10 2019

Magma

f:=Factorial; [0,0] cat [(&+[((-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2) *f(k))): k in [0..n-2]]): n in [2..20]]; // G. C. Greubel, Jul 10 2019

Mathematica

Table[Sum[(n+k+2)!*(-1)^k/(2^(k+2)*(n-k-2)!*k!), {k,0,n-2}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0, 0}, Table[4*n*(n-1)*Pochhammer[1/2, n]*(-2)^(n-2)* Hypergeometric1F1[2-n, -2*n, -2], {n, 2,20}]] (* G. C. Greubel, Aug 14 2017 *)

PARI

for(n=0,20, print1(sum(k=0,n-2, (n+k+2)!*(-1)^k/(2^(k+2)*(n-k-2)!*k!)), ", ")) \\ G. C. Greubel, Aug 14 2017

Sage

f=factorial; [0,0]+[sum((-1)^k*f(n+k+2)/(2^(k+2)*f(n-k-2)*f(k)) for k in (0..n-2)) for n in (2..20)] # G. C. Greubel, Jul 10 2019

Formula

Recurrence: (n-2)*(n-1)*a(n) = -(n-2)*(n+1)*(2*n-1)*a(n-1) + n*(n+1)*a(n-2). - Vaclav Kotesovec, Jul 22 2015

a(n) ~ (-1)^n * 2^(n+1/2) * n^(n+2) / exp(n+1). - Vaclav Kotesovec, Jul 22 2015

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

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

E.g.f.: (1 + 2*x)^(-5/2)*(x*(x + 2)*sqrt(1 + 2*x) + (2*x^3 - 2*x)) * exp(-1 + sqrt(1 + 2*x)). (End)

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