%I #8 Aug 26 2019 21:05:48

%S 1,0,1,1,3,3,8,9,20,26,50,69,125,177,301,440,717,1055,1675,2471,3835,

%T 5660,8627,12697,19095,27978,41581,60650,89244,129490,188925,272676,

%U 394809,566882,815191,1164510,1664295,2365698,3361844,4756030,6723280,9468138,13319299

%N Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

%C First differs from A304967 at a(10) = 50, A304967(10) = 49.

%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 positive integer matrices with only two columns and sum of entries equal to n, up to row and column permutations.

%H Andrew Howroyd, <a href="/A323654/b323654.txt">Table of n, a(n) for n = 0..1000</a>

%F a(2*n) = (A052847(2*n) + A003293(n))/2; a(2*n+1) = A052847(2*n+1)/2. - _Andrew Howroyd_, Aug 26 2019

%e Non-isomorphic representatives of the a(2) = 1 through a(7) = 9 multiset partitions:

%e {{12}} {{122}} {{1122}} {{11222}} {{111222}} {{1112222}}

%e {{1222}} {{12222}} {{112222}} {{1122222}}

%e {{12}{12}} {{12}{122}} {{122222}} {{1222222}}

%e {{112}{122}} {{112}{1222}}

%e {{12}{1122}} {{12}{11222}}

%e {{12}{1222}} {{12}{12222}}

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

%e {{12}{12}{12}} {{122}{1222}}

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

%e Inequivalent representatives of the a(8) = 20 matrices:

%e [4 4] [3 5] [2 6] [1 7]

%e .

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

%e [3 3] [2 4] [1 5] [2 3] [1 4] [2 3] [1 4] [1 3] [2 2] [1 3] [1 3]

%e .

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

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

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

%e .

%e [1 1]

%e [1 1]

%e [1 1]

%e [1 1]

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

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

%Y Cf. A003293, A007716, A052847, A054974, A120733, A321407, A321760, A323655, A323656.

%K nonn

%O 0,5

%A _Gus Wiseman_, Jan 22 2019

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