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%I #29 Sep 07 2020 05:04:06
%S 1,1,2,4,9,21,52,134,361,1009,2926,8768,27121,86373,282864,950866,
%T 3277169,11564353,41739130,153919324,579411641,2224535125,8703993420,
%U 34681783422,140637608089,580019801201,2431509498406,10355296410712
%N A binomial transform of factorial numbers.
%C Binomial transform of A000142 (with interpolated zeros).
%C Row sums of A161556. Hankel transform is A137704. [_Paul Barry_, Apr 11 2010]
%H G. C. Greubel, <a href="/A084261/b084261.txt">Table of n, a(n) for n = 0..880</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 Proposition 4.13 p. 15.
%F a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*k!.
%F a(n) = Sum_{k=0..n} C(n, k)*(k/2)!*((1+(-1)^k)/2) .
%F E.g.f.: exp(x)*(1+sqrt(Pi)/2*x*exp(x^2/4)*erf(x/2)). - _Vladeta Jovovic_, Sep 25 2003
%F 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
%F 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).
%F 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
%t Table[Sum[Binomial[n,2*k]*k!, {k,0,Floor[n/2]}], {n,0,50}] (* _G. C. Greubel_, Jan 24 2017 *)
%o (PARI) for(n=0,50, print1(sum(k=0,floor(n/2), binomial(n,2*k)*k!), ", ")) \\ _G. C. Greubel_, Jan 24 2017
%K easy,nonn
%O 0,3
%A _Paul Barry_, May 26 2003
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