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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

%I #20 Jan 16 2024 22:04:57

%S 1,2,6,9,17,23,36,46,65,80,106,127,161,189,232,268,321,366,430,485,

%T 561,627,716,794,897,988,1106,1211,1345,1465,1616,1752,1921,2074,2262,

%U 2433,2641,2831,3060,3270,3521,3752,4026,4279,4577,4853,5176,5476,5825,6150

%N 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.

%C From _Gus Wiseman_, Jan 22 2019: (Start)

%C 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:

%C {{1}{2}} {{1}{22}} {{1}{122}} {{11}{122}}

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

%C {{12}{12}} {{11}{222}}

%C {{1}{222}} {{12}{122}}

%C {{12}{22}} {{1}{2222}}

%C {{2}{122}} {{12}{222}}

%C {{2}{1122}}

%C {{2}{1222}}

%C {{22}{122}}

%C (End)

%H Colin Barker, <a href="/A054974/b054974.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

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

%F From _Colin Barker_, Jan 16 2017: (Start)

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

%F 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.

%F (End)

%e 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:

%e [0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [0 2] [0 3] [1 1]

%e [1 3] [2 2] [3 1] [4 0] [1 2] [2 1] [3 0] [1 1] [1 2].

%p 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:

%o (PARI) Vec(-x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2) + O(x^60)) \\ _Colin Barker_, Jan 16 2017

%Y Column k=2 of A321615.

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

%K easy,nonn

%O 2,2

%A _Vladeta Jovovic_, May 28 2000

%E More terms from _James A. Sellers_, May 29 2000

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)