A328826 -id:A328826 - cp's OEIS frontend

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

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

Offset: 0

Ralf Stephan, Sep 23 2004

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.

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

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

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

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 *)

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

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

A328921 Triangle read by rows: T(n,k) = number of vertex labeled connected Goldstone diagrams with n interactions and k external potentials.

Original entry on oeis.org

1, 2, 1, 20, 8, 1, 592, 240, 36, 2, 33888, 14208, 2400, 192, 6, 3134208, 1355520, 248640, 24000, 1200, 24, 423974400, 188052480, 36599040, 3978240, 252000, 8640, 120, 78722795520, 35613849600, 7240020480, 853977600, 62657280, 2822400, 70560, 720, 19193652817920, 8816953098240, 1851920179200

Offset: 0

R. J. Mathar, Oct 31 2019

			          1;
          2           1;
         20           8           1;
        592         240          36           2;
      33888       14208        2400         192           6;
    3134208     1355520      248640       24000        1200          24;

R. J. Mathar, Maple program computing A328921 - A328924.
P. J. Rossky and M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, C(n,k)

Cf. A328922, A328923, A328924.

T(n,k) + A328922(n,k) = A328826(n,k).

A328922 Triangle T(n,k): vertex labeled disconnected Goldstone diagrams with n interactions and k external potentials.

Original entry on oeis.org

0, 0, 0, 4, 4, 1, 128, 120, 36, 4, 6432, 5952, 1920, 288, 18, 494592, 458880, 154560, 26400, 2400, 96, 55027200, 51448320, 17832960, 3279360, 352800, 21600, 600, 8455495680, 7975296000, 2819013120, 543110400, 64350720, 4798080, 211680, 4320, 1729137070080, 1644441845760, 589071974400

Offset: 0

R. J. Mathar, Oct 31 2019

			         0;
         0          0;
         4          4          1;
       128        120         36          4;
      6432       5952       1920        288         18;
    494592     458880     154560      26400       2400         96;
  55027200   51448320   17832960    3279360     352800      21600        600;
8455495680 ...

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

Cf. A328826, A328921, A328923, A328924.

T(n,k) + A328921(n,k) = A328826(n,k).

cp's OEIS Frontend

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

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A328921 Triangle read by rows: T(n,k) = number of vertex labeled connected Goldstone diagrams with n interactions and k external potentials.

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A328922 Triangle T(n,k): vertex labeled disconnected Goldstone diagrams with n interactions and k external potentials.

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Author

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A328921 Triangle read by rows: T(n,k) = number of vertex labeled connected Goldstone diagrams with n interactions and k external potentials.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A328922 Triangle T(n,k): vertex labeled disconnected Goldstone diagrams with n interactions and k external potentials.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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