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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

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

Crossrefs

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

Programs

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 *)

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

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