A345132 Number of (n+2) X (n+2) symmetric matrices with nonnegative integer entries, trace 0, with n rows that sum to 2, and 2 rows that sum to 1.

1, 1, 3, 10, 46, 252, 1642, 12316, 104730, 995122, 10450414, 120192924, 1502537932, 20285580880, 294156077364, 4559608340968, 75236088623548, 1316668510772124, 24358939966126900, 475008770990906488, 9737844963832507656, 209366721066736679536

Offset: 0

Author

Stefano Frixione, Jun 30 2021

Keywords

• nonn

Comments

This is the q=1 member of the q-family of sequences F_q(n), defined as the number of (n+2q) X (n+2q) symmetric matrices with nonnegative integer entries, trace 0, with n rows that sum to 2, and 2q rows that sum to 1. It is relevant to the counting of dipole graphs as is discussed in the paper whose link is given below. The q=0 member of this family is the sequence A002137.

Links

• Stefano Frixione and Bryan R. Webber, The role of colour flows in matrix element computations and Monte Carlo simulations, arXiv:2106.13471 [hep-ph], 2021.

Crossrefs

Cf. A002137.

Programs

Mathematica

genF=Exp[-y/2+y^2/4]/Sqrt[1-2*x-y];
(* seq[q,N] gives {F_q(0),...F_q(N)} for any integers q and N *)
seq[q_,N_]:=Table[D[D[genF,{x,q}],{y,n}]/.{x->0,y->0},{n,0,N}]

Formula

E.g.f.: exp(x^2/4-x/2)/(1-x)^(3/2).