A331397 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

1, 2, 14, 128, 1288, 13472, 143840, 1556480, 17006720, 187208192, 2072948224, 23063920640, 257634273280, 2887544053760, 32456082448384, 365710391902208, 4129672996618240, 46721752249794560, 529486122704568320, 6009576477811539968, 68299997524116635648

Offset: 0

Author

Andrew Howroyd, Jan 15 2020

Keywords

• nonn

Comments

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Column k=2 of A331315.

Cf. A011782, A052141, A226994, A331396.

Programs

PARI

seq(n)={Vec(1/(4*sqrt(1 - 12*x + 4*x^2 + O(x*x^n))) + (3 - 4*x)/(4*(1-2*x)))} \\ Andrew Howroyd, Jan 15 2020

Formula

a(n) = (A052141(n) + A011782(n))/2.

G.f.: 1/(4*sqrt(1 - 12*x + 4*x^2)) + (3 - 4*x)/(4*(1-2*x)).

a(n) = A011782(n) * A226994(n).

D-finite with recurrence n*a(n) +2*(-8*n+5)*a(n-1) +28*(2*n-3)*a(n-2) +8*(-8*n+19)*a(n-3) +16*(n-3)*a(n-4)=0. - R. J. Mathar, Mar 13 2023