%I #43 Nov 22 2021 14:58:36

%S 1,3,38,1158,65304,5900520,780827760,142358474160,34209760152960,

%T 10478436416945280,3984884716852972800,1842169367191937414400,

%U 1017403495472574045158400,661599650478455071589606400,500354503197888042597961267200,435447353708763072625260119808000

%N a(n) = Sum_{k=0..n} C(n,k)*(2*n-k)!.

%C Diagonal of Euler-Seidel matrix with start sequence n!.

%C Number of ways to use the elements of {1,..,k}, n<=k<=2n, once each to form a sequence of n lists, each having length 1 or 2. - Bob Proctor, Apr 18 2005, Jun 26 2006

%C Replace "lists" by "sets": A105749.

%H Robert Israel, <a href="/A099022/b099022.txt">Table of n, a(n) for n = 0..224</a>

%H L. I. Nicolaescu, <a href="http://nyjm.albany.edu/j/2004/10-7.pdf">Derangements and asymptotics of the Laplace transforms of large powers of a polynomial</a>, New York J. Math. 10 (2004) 117-131.

%H Robert A. Proctor, <a href="http://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way For Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.

%H P. J. Rossky, M. Karplus, <a href="https://doi.org/10.1063/1.432387">The enumeration of Goldstone diagrams in many-body perturbation theory</a>, J. Chem. Phys. 64 (1976) 1569, equation (9).

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F T(2*n, n), where T is the triangle in A076571.

%F a(n) = n!*A001517(n).

%F A082765(n) = Sum[C(n, k)*a(k), 0<=k<=n].

%F a(n) = 2*n*(2*n-1)*a(n-1)+n*(n-1)*a(n-2). - _Vladeta Jovovic_, Sep 27 2004

%F a(n) = int {x = 0..inf} exp(-x)*(x + x^2)^n dx. Applying the results of Nicolaescu, Section 3.2 to this integral we obtain the asymptotic expansion a(n) ~ (2*n)!*exp(1/2)*( 1 - 1/(16*n) - 191/(6144*n^2) + O(1/n^3) ). - _Peter Bala_, Jul 07 2014

%p f:= gfun:-rectoproc({a(n)=2*n*(2*n-1)*a(n-1)+n*(n-1)*a(n-2), a(0)=1,a(1)=3},a(n),remember):

%p map(f, [$0..20]); # _Robert Israel_, Feb 15 2017

%t Table[(2k)! Hypergeometric1F1[-k, -2k, 1], {k, 0, 10}] (* _Vladimir Reshetnikov_, Feb 16 2011 *)

%t Table[Sum[Binomial[n,k](2n-k)!,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Nov 22 2021 *)

%o (PARI) for(n=0,25, print1(sum(k=0,n, binomial(n,k)*(2*n-k)!), ", ")) \\ _G. C. Greubel_, Dec 31 2017

%Y Cf. A001517, A076571, A082765 (binomial transform), A105749, row sums of A328826.

%K nonn,easy

%O 0,2

%A _Ralf Stephan_, Sep 23 2004