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

2,2

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)

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).

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)

Example

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].

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:

Prog

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

Crossrefs

Column k=2 of A321615.

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

Sequence in context: A280228 A254057 A257083 * A072481 A032471 A358258

Adjacent sequences: A054971 A054972 A054973 * A054975 A054976 A054977

Keyword

easy,nonn

Author

Vladeta Jovovic, May 28 2000

Extensions

More terms from James A. Sellers, May 29 2000

Status

approved