OFFSET
0,3
COMMENTS
Essentially the same as A072374. - R. J. Mathar, Jun 18 2008
Diagonal sums of A008279. - Paul Barry, Feb 11 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jonathan Fang, Zachary Hamaker, and Justin Troyka, On pattern avoidance in matchings and involutions, arXiv:2009.00079 [math.CO], 2020. See Theorem 1.6 (b).
Guo-Niu Han, Hankel Continued fractions and Hankel determinants of the Euler numbers, arXiv:1906.00103 [math.CO], 2019. See p. 27.
Qiong Qiong Pan and Jiang Zeng, The gamma-coefficients of Branden's (p,q)-Eulerian polynomials and André permutations, arXiv:1910.01747 [math.CO], 2019.
FORMULA
a(n) = Sum{k=0..n} C(k,n-k)*(n-k)!.
From Paul Barry, Feb 11 2009: (Start)
G.f.: 1/(1-x-x^2/(1-x^2/(1-x-2x^2/(1-2x^2/(1-x-3x^2/(1-3x^2/(1-x-4x^2/(1-4x^2/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*k!. (End)
D-finite with recurrence -2*a(n) + 3*a(n-1) + (n-1)*a(n-2) + (-n+1)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012. Proof in [Han 2019]
a(n) ~ sqrt(Pi) * exp(sqrt(n/2) - n/2 + 1/8) * n^((n+1)/2) / 2^(n/2+1) * (1 + 37/(48*sqrt(2*n))). - Vaclav Kotesovec, Feb 08 2014
a(n) = (a(n-1) + n * a(n-2) + 1)/2 for n > 1. - Seiichi Manyama, Nov 19 2022
MATHEMATICA
Table[Sum[Binomial[n-k, k]*k!, {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 08 2014 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(k, n-k)*(n-k)!); \\ Michel Marcus, Sep 02 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 14 2006
EXTENSIONS
More terms from Vaclav Kotesovec, Jun 04 2019
STATUS
approved