%I #7 Jan 28 2024 18:10:51
%S 1,0,0,1,0,1,0,2,3,0,2,4,0,4,15,8,0,4,24,19,0,7,60,79,23,0,8,101,213,
%T 84,0,12,210,615,424,66,0,14,357,1523,1533,363,0,21,679,3851,5580,
%U 2217,212,0,24,1142,8963,17836,10379,1575,0,34,2049,20840,55730,45866,11616,686
%N Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no singletons or vertices that appear only once, 0 <= k <= floor(n/2).
%C T(n,k) is the number of nonnegative integer matrices with sum of values n, k rows and every row and column sum at least two up to permutation of rows and columns.
%H Andrew Howroyd, <a href="/A369287/b369287.txt">Table of n, a(n) for n = 0..675</a> (rows 0..50)
%F T(2*n, n) = A050535(n).
%e Triangle begins:
%e 1;
%e 0;
%e 0, 1;
%e 0, 1;
%e 0, 2, 3;
%e 0, 2, 4;
%e 0, 4, 15, 8;
%e 0, 4, 24, 19;
%e 0, 7, 60, 79, 23;
%e 0, 8, 101, 213, 84;
%e 0, 12, 210, 615, 424, 66;
%e 0, 14, 357, 1523, 1533, 363;
%e 0, 21, 679, 3851, 5580, 2217, 212;
%e 0, 24, 1142, 8963, 17836, 10379, 1575;
%e ...
%e The T(5,1) = 2 multiset partitions are:
%e {{1,1,1,1,1}},
%e {{1,1,1,2,2}}.
%e The corresponding T(5,1) = 2 matrices are:
%e [5] [3 2].
%e The T(5,2) = 4 matrices are:
%e [3] [3 0] [2 1] [2 1]
%e [2] [0 2] [1 1] [0 2],
%o (PARI)
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
%o H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); (1 - x)*x*Ser(K(q,t,k)) - c*x}
%o G(n,y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}
%o T(n)={my(v=Vec(G(n,'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
%o { my(A=T(15)); for(i=1, #A, print(A[i])) }
%Y Row sums are A320665.
%Y Columns k=0..1 are A000007, A002865(n>0).
%Y Cf. A050535, A369286.
%K nonn,tabf
%O 0,8
%A _Andrew Howroyd_, Jan 28 2024