A328826 Triangle read by rows: binomial(n,k)*(2*n-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

Author

R. J. Mathar, Oct 28 2019

Keywords

• nonn
• easy
• tabl

Comments

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

Examples

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

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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

Formula

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

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