A081124 Binomial transform of floor(n/2)!.

1, 2, 4, 8, 17, 38, 90, 224, 585, 1594, 4520, 13288, 40409, 126782, 409646, 1360512, 4637681, 16202034, 57941164, 211860488, 791272129, 3015807254, 11719800674, 46401584096, 187039192185, 767058993386, 3198568491792, 13553864902504

Offset: 0

Author

Paul Barry, Mar 07 2003

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..880
• Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012

Crossrefs

Cf. A081123, A004526.

Cf. A084261.

Programs

Mathematica

Table[Sum[Binomial[n,k]*Floor[k/2]!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 15 2013 *)

PARI

for(n=0,50, print1(sum(k=0,n, binomial(n,k)*(floor(k/2))!), ", ")) \\ G. C. Greubel, Feb 02 2017

Formula

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

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

Conjecture: 2*a(n) -4*a(n-1) +(-n+2)*a(n-2) +(n-1)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ sqrt(Pi*n)/2 * exp(sqrt(2*n)-n/2-1/2)*(n/2)^(n/2) * (1+5/(3*sqrt(2*n))). - Vaclav Kotesovec, Aug 15 2013