A328826 - OEIS

%I #10 Feb 03 2022 19:30:19

%S 1,2,1,24,12,2,720,360,72,6,40320,20160,4320,480,24,3628800,1814400,

%T 403200,50400,3600,120,479001600,239500800,54432000,7257600,604800,

%U 30240,720,87178291200,43589145600,10059033600,1397088000,127008000,7620480,282240,5040,20922789888000

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

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

%H P. J. Rossky, M. Karplus, <a href="https://doi.org/10.1063/1.432387">The enumeration of Goldstone diagrams in many-body perturbation theory</a>, J. Chem. Phys. 64 (1976) 1569, equation (9) and Table 1.

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

%F T(n,k) = A328921(n,k) + A328922(n,k). - _R. J. Mathar_, Nov 02 2019

%e The triangle starts

%e     1;

%e     2     1;

%e    24    12     2;

%e   720   360    72     6;

%e 40320 20160  4320   480    24;

%p A328826 := proc(n,k)

%p         binomial(n,k)*(2*n-k)! ;

%p end proc:

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

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

%K nonn,easy,tabl

%O 0,2

%A _R. J. Mathar_, Oct 28 2019