A006199 Bessel polynomial {y_n}'(-1). (Formerly M3082)

0, 1, -3, 21, -185, 2010, -25914, 386407, -6539679, 123823305, -2593076255, 59505341676, -1484818160748, 40025880386401, -1159156815431055, 35891098374564105, -1183172853341759129, 41372997479943753582, -1529550505546305534414, 59608871544962952539335

Offset

1,3

Comments

Absolute values give partitions into pairs.

References

G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 1..400

Index entries for sequences related to Bessel functions or polynomials

Formula

a(n) = A000806(n) + (n-1) * A000806(n-1). - Sean A. Irvine, Jan 23 2017

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

a(n) = 2*n*(1/2)_{n} * (-2)^(n-1) * hyergeometric1f1(1-n; -2*n; -2), where (a)_{n} is the Pochhammer symbol.

E.g.f.: (1+2*x)^(-3/2)*( (1+2*x)^(3/2) - x*(1+2*x)^(1/2) - x -1) * exp(sqrt(1+2*x) - 1), for offset 0. (End)

G.f.: (x/(1-x)^3)*hypergeometric2f0(2,3/2; - ; -2*x/(1-x)^2), for offset 0. - G. C. Greubel, Aug 16 2017

Mathematica

Join[{0}, Table[2*n*Pochhammer[1/2, n]*(-2)^(n - 1)* Hypergeometric1F1[1 - n, -2*n, -2], {n, 1, 50}]] (* G. C. Greubel, Aug 14 2017 *)

Prog

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

Crossrefs

Cf. A000806, A001514, A065707, A065920, A065921, A065922.

Sequence in context: A054879 A333090 A131763 * A083063 A012163 A012055

Adjacent sequences: A006196 A006197 A006198 * A006200 A006201 A006202

Keyword

sign

Author

N. J. A. Sloane

Status

approved