A065920 Bessel polynomial {y_n}'(2).

0, 1, 15, 246, 4810, 111315, 2994621, 92069740, 3188772756, 122934101445, 5223324100555, 242563221769506, 12224586738476190, 664572113979550231, 38767776344788218105, 2415639337342677314520, 160131212043826343202856

Offset

0,3

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

Formula

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

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

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) + 8*x - 1)*exp((1 - sqrt(1 - 4*x))/2)/(4*(1 - 4*x)^(3/2)). (End)

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

Mathematica

Table[Sum[(n+k+1)!/(2*(n-k-1)!*k!), {k, 0, n-1}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *)
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 *)

Prog

(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

Crossrefs

Cf. A001514, A065921, A065922.

Sequence in context: A059615 A215855 A163031 * A273624 A223424 A218806

Adjacent sequences: A065917 A065918 A065919 * A065921 A065922 A065923

Keyword

nonn

Author

N. J. A. Sloane, Dec 08 2001

Status

approved