A084261 A binomial transform of factorial numbers.

1, 1, 2, 4, 9, 21, 52, 134, 361, 1009, 2926, 8768, 27121, 86373, 282864, 950866, 3277169, 11564353, 41739130, 153919324, 579411641, 2224535125, 8703993420, 34681783422, 140637608089, 580019801201, 2431509498406, 10355296410712

Offset

0,3

Comments

Binomial transform of A000142 (with interpolated zeros).

Row sums of A161556. Hankel transform is A137704. [Paul Barry, Apr 11 2010]

Links

G. C. Greubel, Table of n, a(n) for n = 0..880

Jonathan Fang, Zachary Hamaker, and Justin Troyka, On pattern avoidance in matchings and involutions, arXiv:2009.00079 [math.CO], 2020. See Proposition 4.13 p. 15.

Formula

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*k!.

a(n) = Sum_{k=0..n} C(n, k)*(k/2)!*((1+(-1)^k)/2) .

E.g.f.: exp(x)*(1+sqrt(Pi)/2*x*exp(x^2/4)*erf(x/2)). - Vladeta Jovovic, Sep 25 2003

O.g.f.: A(x) = 1/(1-x-x^2/(1-x-x^2/(1-x-2*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-[(n+1)/2]*x^2/(1- ...))))))) (continued fraction). - Paul D. Hanna, Jan 17 2006

a_n ~ (1/2) * sqrt(Pi*n/e)*(n/2)^(n/2)*exp(-n/2 + sqrt(2n)). - Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006: (cf. A002896).

Conjecture: 2*a(n) -4*a(n-1) +(-n+2)*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012

Mathematica

Table[Sum[Binomial[n, 2*k]*k!, {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Jan 24 2017 *)

Prog

(PARI) for(n=0, 50, print1(sum(k=0, floor(n/2), binomial(n, 2*k)*k!), ", ")) \\ G. C. Greubel, Jan 24 2017

Crossrefs

Sequence in context: A195980 A136753 A289666 * A063026 A106219 A198304

Adjacent sequences: A084258 A084259 A084260 * A084262 A084263 A084264

Keyword

easy,nonn

Author

Paul Barry, May 26 2003

Status

approved