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%I #48 Feb 15 2023 09:40:24
%S 1,1,2,3,6,11,24,51,122,291,756,1979,5526,15627,46496,140451,442194,
%T 1414931,4687212,15785451,54764846,193129659,698978136,2570480147,
%U 9672977706,36967490691,144232455524,571177352091,2304843053382,9434493132011,39289892366736
%N Row sums of number triangle A122851.
%C Essentially the same as A072374. - _R. J. Mathar_, Jun 18 2008
%C Diagonal sums of A008279. - _Paul Barry_, Feb 11 2009
%H Vincenzo Librandi, <a href="/A122852/b122852.txt">Table of n, a(n) for n = 0..200</a>
%H Jonathan Fang, Zachary Hamaker, and Justin Troyka, <a href="https://arxiv.org/abs/2009.00079">On pattern avoidance in matchings and involutions</a>, arXiv:2009.00079 [math.CO], 2020. See Theorem 1.6 (b).
%H Guo-Niu Han, <a href="https://arxiv.org/abs/1906.00103">Hankel Continued fractions and Hankel determinants of the Euler numbers</a>, arXiv:1906.00103 [math.CO], 2019. See p. 27.
%H Qiong Qiong Pan and Jiang Zeng, <a href="https://arxiv.org/abs/1910.01747">The gamma-coefficients of Branden's (p,q)-Eulerian polynomials and André permutations</a>, arXiv:1910.01747 [math.CO], 2019.
%F a(n) = Sum{k=0..n} C(k,n-k)*(n-k)!.
%F From _Paul Barry_, Feb 11 2009: (Start)
%F 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).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*k!. (End)
%F 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]
%F 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
%F a(n) = (a(n-1) + n * a(n-2) + 1)/2 for n > 1. - _Seiichi Manyama_, Nov 19 2022
%t Table[Sum[Binomial[n-k,k]*k!,{k,0,Floor[n/2]}],{n,0,20}] (* _Vaclav Kotesovec_, Feb 08 2014 *)
%o (PARI) a(n) = sum(k=0, n, binomial(k,n-k)*(n-k)!); \\ _Michel Marcus_, Sep 02 2020
%Y Cf. A072374, A008279.
%Y Cf. A357532, A357533, A357570.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Sep 14 2006
%E More terms from _Vaclav Kotesovec_, Jun 04 2019
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