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

0,3

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

Table of n, a(n) for n=0..21.

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.

Formula

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

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}]

Crossrefs

Cf. A002137.

Sequence in context: A074508 A143500 A058112 * A020008 A167999 A000849

Adjacent sequences: A345129 A345130 A345131 * A345133 A345134 A345135

Keyword

nonn

Author

Stefano Frixione, Jun 30 2021

Status

approved