A054974 Number of nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.

1, 2, 6, 9, 17, 23, 36, 46, 65, 80, 106, 127, 161, 189, 232, 268, 321, 366, 430, 485, 561, 627, 716, 794, 897, 988, 1106, 1211, 1345, 1465, 1616, 1752, 1921, 2074, 2262, 2433, 2641, 2831, 3060, 3270, 3521, 3752, 4026, 4279, 4577, 4853, 5176, 5476, 5825, 6150

Offset: 2

Author

Vladeta Jovovic, May 28 2000

Keywords

• easy
• nonn

Comments

From Gus Wiseman, Jan 22 2019: (Start)

Also the number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices and exactly 2 (not necessarily distinct) edges. For example, non-isomorphic representatives of the a(2) = 1 through a(5) = 9 multiset partitions are:

{{1}{2}} {{1}{22}} {{1}{122}} {{11}{122}}

{{2}{12}} {{11}{22}} {{1}{1222}}

{{12}{12}} {{11}{222}}

{{1}{222}} {{12}{122}}

{{12}{22}} {{1}{2222}}

{{2}{122}} {{12}{222}}

{{2}{1122}}

{{2}{1222}}

{{22}{122}}

(End)

Examples

			There are 9 nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to 5, up to row and column permutation:
[0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [0 2] [0 3] [1 1]
[1 3] [2 2] [3 1] [4 0] [1 2] [2 1] [3 0] [1 1] [1 2].
		

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Column k=2 of A321615.

Cf. A007716, A052847, A053307, A323654, A323655, A323656.

Programs

Maple

gf := -x^2*(x^3-x^2-1)/((x^2-1)^2*(x-1)^2): s := series(gf, x, 101): for i from 2 to 100 do printf(`%d,`,coeff(s,x,i)) od:

PARI

Vec(-x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2) + O(x^60)) \\ Colin Barker, Jan 16 2017

Formula

G.f.: -x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2).

From Colin Barker, Jan 16 2017: (Start)

a(n) = (6 - 6*(-1)^n + (9*(-1)^n-17)*n + 12*n^2 + 2*n^3) / 48.

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>7.

(End)

Extensions

More terms from James Sellers, May 29 2000