%I #8 Aug 26 2019 21:44:48

%S 0,0,2,4,14,28,69,134,285,536,1050,1918,3566,6346,11363,19771,34405,

%T 58677,99797,167223,279032,460264,755560,1228849,1988680,3193513,

%U 5103104,8100712,12798207,20102883,31434374,48900337,75746745,116787611,179342230,274238159

%N Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

%C Also the number of nonnegative integer matrices with only two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

%H Andrew Howroyd, <a href="/A323656/b323656.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A323655(n) - A000041(n). - _Andrew Howroyd_, Aug 26 2019

%e Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 distinct vertices:

%e {{12}} {{122}} {{1122}}

%e {{1}{2}} {{1}{22}} {{1222}}

%e {{2}{12}} {{1}{122}}

%e {{1}{2}{2}} {{11}{22}}

%e {{12}{12}}

%e {{1}{222}}

%e {{12}{22}}

%e {{2}{122}}

%e {{1}{1}{22}}

%e {{1}{2}{12}}

%e {{1}{2}{22}}

%e {{2}{2}{12}}

%e {{1}{1}{2}{2}}

%e {{1}{2}{2}{2}}

%e Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 edges:

%e {{1}{1}} {{1}{11}} {{1}{111}}

%e {{1}{2}} {{1}{22}} {{11}{11}}

%e {{1}{23}} {{1}{122}}

%e {{2}{12}} {{11}{22}}

%e {{12}{12}}

%e {{1}{222}}

%e {{12}{22}}

%e {{1}{233}}

%e {{12}{33}}

%e {{1}{234}}

%e {{12}{34}}

%e {{13}{23}}

%e {{2}{122}}

%e {{3}{123}}

%e Inequivalent representatives of the a(4) = 14 matrices:

%e [2 2] [1 3]

%e .

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

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

%e .

%e [1 0] [1 0] [1 0] [0 1]

%e [1 0] [0 1] [0 1] [0 1]

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

%e .

%e [1 0] [1 0]

%e [1 0] [0 1]

%e [0 1] [0 1]

%e [0 1] [0 1]

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o seq(n)={concat(0, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2 - EulerT(vector(n,k,1)))} \\ _Andrew Howroyd_, Aug 26 2019

%Y Cf. A000041, A007716, A052847, A054974, A120733, A316980, A323654, A323655, A323656.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jan 22 2019

%E Terms a(11) and beyond from _Andrew Howroyd_, Aug 26 2019