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

Author

Paul Barry, May 26 2003

Keywords

• easy
• nonn

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.

Programs

Mathematica

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

PARI

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

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