A065922 Bessel polynomial {y_n}'(4).

0, 1, 27, 846, 32290, 1472535, 78444261, 4789283212, 329976556596, 25336918039005, 2145912573891295, 198763621138900026, 19988975122377164982, 2169175884299414423251, 252661578519463668740745, 31442098485128401965118680, 4163361054820272025075769896

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..330
• Index entries for sequences related to Bessel functions or polynomials

Crossrefs

Cf. A001514, A065920, A065921.

Programs

Mathematica

Table[Sum[(n+k+1)!*2^(k-1)/((n-k-1)!*k!), {k,0,n-1}], {n,0,20}] (* Vaclav Kotesovec, Jul 22 2015 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==((2n-1)^3 a[n-1]+n^2 a[n-2])/ (n-1)^2}, a,{n,20}] (* Harvey P. Dale, May 23 2016 *)

PARI

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

Formula

Recurrence: (n-1)^2*a(n) = (2*n - 1)^3*a(n-1) + n^2*a(n-2). - Vaclav Kotesovec, Jul 22 2015

a(n) ~ 2^(3*n-3/2) * n^(n+1) / exp(n-1/4). - Vaclav Kotesovec, Jul 22 2015

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

a(n) = 2*n*(1/2)_{n}*8^(n - 1)* hypergeometric1f1(1 - n, -2*n, 1/2).

E.g.f.: ((1 - 8*x)^(3/2) + 4*x*(1 - 8*x)^(1/2) + 24*x - 1)* exp((1 - sqrt(1 - 8*x))/4)/(16*(1 - 8*x)^(3/2)). (End)

G.f.: (t/(1-t)^3)*hypergeometric2f0(2,3/2; - ; 8*t/(1-t)^2). - G. C. Greubel, Aug 16 2017