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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.
8
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)
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.
Sequence in context: A280228 A254057 A257083 * A072481 A032471 A358258
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, May 28 2000
EXTENSIONS
More terms from James A. Sellers, May 29 2000
STATUS
approved