A328921 -id:A328921 - cp's OEIS frontend

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

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

A328923 Triangle T(n,k) read by rows: connected topologies of Goldstone diagrams with n interactions and k external potentials.

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 76, 60, 18, 2, 2124, 1776, 600, 96, 6, 97968, 84720, 31080, 6000, 600, 24, 6624720, 5876640, 2287440, 497280, 63000, 4320, 120, 615022560, 556466400, 226250640, 53373600, 7832160, 705600, 35280, 720, 74975211360, 68882446080, 28936252800, 7240020480

Offset: 0

R. J. Mathar, Oct 31 2019

			         1;
         2          1;
         6          4          1;
        76         60         18          2;
      2124       1776        600         96          6;
     97968      84720      31080       6000        600         24;
   6624720    5876640    2287440     497280      63000       4320        120;

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

Cf. A328921-A328924

T(n,0) = A328921(n,0) / 2^n +(n-1)! .

T(n,k) = 2^(k-n)*A328921(n,k), k>0 .

A328924 Triangle T(n,k) read by rows: connected topologies of the effective potential in Goldstone diagrams with n interactions and k external potentials.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 12, 12, 6, 2, 300, 336, 168, 48, 6, 13680, 15120, 7560, 2160, 360, 24, 920400, 1000800, 498960, 146880, 27000, 2880, 120, 85004640, 91314720, 45516240, 13698720, 2691360, 342720, 25200, 720, 10314511200, 10981474560, 5476625280, 1675699200, 344433600, 48545280, 4515840, 241920, 5040

Offset: 0

R. J. Mathar, Oct 31 2019

			         1;
         0          1;
         2          0          1;
        12         12          6          2;
       300        336        168         48          6;
     13680      15120       7560       2160        360         24;
    920400    1000800     498960     146880      27000       2880        120;
  85004640   91314720   45516240   13698720    2691360     342720      25200        720;

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

Cf. A328921-A328923.

T(n,k) = sum_{t=k..n} (-1)^(k+t) *2^(t-k) *binomial(t,k) *A328923(n,t).

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

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

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A328923 Triangle T(n,k) read by rows: connected topologies of Goldstone diagrams with n interactions and k external potentials.

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A328924 Triangle T(n,k) read by rows: connected topologies of the effective potential in 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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cp's OEIS Frontend

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

Views

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Maple

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A328923 Triangle T(n,k) read by rows: connected topologies of Goldstone diagrams with n interactions and k external potentials.

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Formula

A328924 Triangle T(n,k) read by rows: connected topologies of the effective potential in 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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A328826 Triangle read by rows: binomial(n,k)(2n-k)!, n>=0, 0<=k<=n.