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A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

%I A323654 #8 Aug 26 2019 21:05:48
%S A323654 1,0,1,1,3,3,8,9,20,26,50,69,125,177,301,440,717,1055,1675,2471,3835,
%T A323654 5660,8627,12697,19095,27978,41581,60650,89244,129490,188925,272676,
%U A323654 394809,566882,815191,1164510,1664295,2365698,3361844,4756030,6723280,9468138,13319299
%N A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.
%C A323654 First differs from A304967 at a(10) = 50, A304967(10) = 49.
%C A323654 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 A323654 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 A323654 Andrew Howroyd, <a href="/A323654/b323654.txt">Table of n, a(n) for n = 0..1000</a>
%F A323654 a(2*n) = (A052847(2*n) + A003293(n))/2; a(2*n+1) = A052847(2*n+1)/2. - _Andrew Howroyd_, Aug 26 2019
%e A323654 Non-isomorphic representatives of the a(2) = 1 through a(7) = 9 multiset partitions:
%e A323654   {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}      {{1112222}}
%e A323654                    {{1222}}    {{12222}}    {{112222}}      {{1122222}}
%e A323654                    {{12}{12}}  {{12}{122}}  {{122222}}      {{1222222}}
%e A323654                                             {{112}{122}}    {{112}{1222}}
%e A323654                                             {{12}{1122}}    {{12}{11222}}
%e A323654                                             {{12}{1222}}    {{12}{12222}}
%e A323654                                             {{122}{122}}    {{122}{1122}}
%e A323654                                             {{12}{12}{12}}  {{122}{1222}}
%e A323654                                                             {{12}{12}{122}}
%e A323654 Inequivalent representatives of the a(8) = 20 matrices:
%e A323654   [4 4] [3 5] [2 6] [1 7]
%e A323654 .
%e A323654   [1 1] [1 1] [1 1] [2 1] [2 1] [1 2] [1 2] [3 1] [2 2] [2 2] [1 3]
%e A323654   [3 3] [2 4] [1 5] [2 3] [1 4] [2 3] [1 4] [1 3] [2 2] [1 3] [1 3]
%e A323654 .
%e A323654   [1 1] [1 1] [1 1] [1 1]
%e A323654   [1 1] [1 1] [2 1] [1 2]
%e A323654   [2 2] [1 3] [1 2] [1 2]
%e A323654 .
%e A323654   [1 1]
%e A323654   [1 1]
%e A323654   [1 1]
%e A323654   [1 1]
%o A323654 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A323654 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 A323654 Cf. A003293, A007716, A052847, A054974, A120733, A321407, A321760, A323655, A323656.
%K A323654 nonn
%O A323654 0,5
%A A323654 _Gus Wiseman_, Jan 22 2019
%E A323654 Terms a(11) and beyond from _Andrew Howroyd_, Aug 26 2019