A328922 -id:A328922 - cp's OEIS frontend

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

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).

A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 24, 12, 2, 720, 360, 72, 6, 40320, 20160, 4320, 480, 24, 3628800, 1814400, 403200, 50400, 3600, 120, 479001600, 239500800, 54432000, 7257600, 604800, 30240, 720, 87178291200, 43589145600, 10059033600, 1397088000, 127008000, 7620480, 282240, 5040, 20922789888000

Offset: 0

R. J. Mathar, Oct 28 2019

Vertex-labeled disconnected Goldstone diagrams with n vertices and k single-particle potentials.

			The triangle starts
    1;
    2     1;
   24    12     2;
  720   360    72     6;
40320 20160  4320   480    24;

P. J. Rossky, M. Karplus, The enumeration of Goldstone diagrams in many-body perturbation theory, J. Chem. Phys. 64 (1976) 1569, equation (9) and Table 1.

Cf. A099022 (row sums), A000142 (diagonal), A010050 (column k=0), A002674 (k=1).

A328826 := proc(n,k)
        binomial(n,k)*(2*n-k)! ;
end proc:

Table[Binomial[n,k](2n-k)!,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Feb 03 2022 *)

T(n,k)= binomial(n,k)*(2*n-k)!.

T(n,k) = A328921(n,k) + A328922(n,k). - R. J. Mathar, Nov 02 2019

cp's OEIS Frontend

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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A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.

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cp's OEIS Frontend

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

A328826 Triangle read by rows: binomial(n,k)*(2*n-k)!, n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.