This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A054974 #21 Jul 02 2025 16:01:59
%S A054974 1,2,6,9,17,23,36,46,65,80,106,127,161,189,232,268,321,366,430,485,
%T A054974 561,627,716,794,897,988,1106,1211,1345,1465,1616,1752,1921,2074,2262,
%U A054974 2433,2641,2831,3060,3270,3521,3752,4026,4279,4577,4853,5176,5476,5825,6150
%N 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.
%C A054974 From _Gus Wiseman_, Jan 22 2019: (Start)
%C A054974 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 A054974 {{1}{2}} {{1}{22}} {{1}{122}} {{11}{122}}
%C A054974 {{2}{12}} {{11}{22}} {{1}{1222}}
%C A054974 {{12}{12}} {{11}{222}}
%C A054974 {{1}{222}} {{12}{122}}
%C A054974 {{12}{22}} {{1}{2222}}
%C A054974 {{2}{122}} {{12}{222}}
%C A054974 {{2}{1122}}
%C A054974 {{2}{1222}}
%C A054974 {{22}{122}}
%C A054974 (End)
%H A054974 Colin Barker, <a href="/A054974/b054974.txt">Table of n, a(n) for n = 2..1000</a>
%H A054974 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F A054974 G.f.: -x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2).
%F A054974 From _Colin Barker_, Jan 16 2017: (Start)
%F A054974 a(n) = (6 - 6*(-1)^n + (9*(-1)^n-17)*n + 12*n^2 + 2*n^3) / 48.
%F A054974 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 A054974 (End)
%e A054974 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 A054974 [0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [0 2] [0 3] [1 1]
%e A054974 [1 3] [2 2] [3 1] [4 0] [1 2] [2 1] [3 0] [1 1] [1 2].
%p A054974 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 A054974 (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 A054974 Column k=2 of A321615.
%Y A054974 Cf. A007716, A052847, A053307, A323654, A323655, A323656.
%K A054974 easy,nonn
%O A054974 2,2
%A A054974 _Vladeta Jovovic_, May 28 2000
%E A054974 More terms from _James Sellers_, May 29 2000