A099022 a(n) = Sum_{k=0..n} C(n,k)*(2*n-k)!.

1, 3, 38, 1158, 65304, 5900520, 780827760, 142358474160, 34209760152960, 10478436416945280, 3984884716852972800, 1842169367191937414400, 1017403495472574045158400, 661599650478455071589606400, 500354503197888042597961267200, 435447353708763072625260119808000

Offset

0,2

Comments

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

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

Replace "lists" by "sets": A105749.

Links

Robert Israel, Table of n, a(n) for n = 0..224

L. I. Nicolaescu, Derangements and asymptotics of the Laplace transforms of large powers of a polynomial, New York J. Math. 10 (2004) 117-131.

Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.

P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (9).

Index entries for related partition-counting sequences

Formula

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

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

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

a(n) = 2*n*(2*n-1)*a(n-1)+n*(n-1)*a(n-2). - Vladeta Jovovic, Sep 27 2004

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

Maple

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):
map(f, [$0..20]); # Robert Israel, Feb 15 2017

Mathematica

Table[(2k)! Hypergeometric1F1[-k, -2k, 1], {k, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)
Table[Sum[Binomial[n, k](2n-k)!, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Nov 22 2021 *)

Prog

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

Crossrefs

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

Sequence in context: A335529 A062155 A278927 * A229365 A136638 A213002

Adjacent sequences: A099019 A099020 A099021 * A099023 A099024 A099025

Keyword

nonn,easy

Author

Ralf Stephan, Sep 23 2004

Status

approved