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 A122852 Row sums of number triangle A122851. 9
 1, 1, 2, 3, 6, 11, 24, 51, 122, 291, 756, 1979, 5526, 15627, 46496, 140451, 442194, 1414931, 4687212, 15785451, 54764846, 193129659, 698978136, 2570480147, 9672977706, 36967490691, 144232455524, 571177352091, 2304843053382, 9434493132011, 39289892366736 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A072374, A008279. Cf. A357532, A357533, A357570. Sequence in context: A047750 A072187 A072374 * A192573 A284994 A107113 Adjacent sequences: A122849 A122850 A122851 * A122853 A122854 A122855 KEYWORD easy,nonn AUTHOR Paul Barry, Sep 14 2006 EXTENSIONS More terms from Vaclav Kotesovec, Jun 04 2019 STATUS approved

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