A065921 Bessel polynomial {y_n}'(3).

0, 1, 21, 501, 14455, 496770, 19911486, 913839031, 47303189361, 2727741976785, 173455231572865, 12060173714421756, 910301022642409476, 74134150415555474881, 6479678618270868170265, 605042444997867941987385, 60110944381660549838273911

Offset

0,3

References

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

G. C. Greubel, Table of n, a(n) for n = 0..345

Index entries for sequences related to Bessel functions or polynomials

Formula

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

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

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

a(n) = 2*n*(1/2)_{n}*6^(n - 1)* hypergeometric1f1(1 - n, -2*n, 2/3).

E.g.f.: ((1 - 6*x)^(3/2) + 3*x*(1 - 6*x)^(1/2) + 15*x - 1) * exp((1 - sqrt(1 - 6*x))/3)/(9*(1 - 6*x)^(3/2)). (End)

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

Mathematica

Table[Sum[(n+k+1)!*3^k/((n-k-1)!*k!*2^(k+1)), {k, 0, n-1}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 22 2015 *)
Join[{0}, Table[2*n*Pochhammer[1/2, n]*6^(n - 1)* Hypergeometric1F1[1 - n, -2*n, 2/3], {n, 1, 50}]] (* G. C. Greubel, Aug 14 2017 *)

Prog

(PARI) for(n=0, 50, print1(sum(k=0, n-1, (n+k+1)!*3^k/((n-k-1)!*k! *2^(k+1))), ", ")) \\ G. C. Greubel, Aug 14 2017

Crossrefs

Cf. A001514, A065920, A065922.

Sequence in context: A015695 A006299 A231542 * A129993 A359852 A209348

Adjacent sequences: A065918 A065919 A065920 * A065922 A065923 A065924

Keyword

nonn

Author

N. J. A. Sloane, Dec 08 2001

Status

approved