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A369287 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).

%I A369287 #7 Jan 28 2024 18:10:51
%S A369287 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 A369287 84,0,12,210,615,424,66,0,14,357,1523,1533,363,0,21,679,3851,5580,
%U A369287 2217,212,0,24,1142,8963,17836,10379,1575,0,34,2049,20840,55730,45866,11616,686
%N A369287 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 A369287 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 A369287 Andrew Howroyd, <a href="/A369287/b369287.txt">Table of n, a(n) for n = 0..675</a> (rows 0..50)
%F A369287 T(2*n, n) = A050535(n).
%e A369287 Triangle begins:
%e A369287   1;
%e A369287   0;
%e A369287   0,  1;
%e A369287   0,  1;
%e A369287   0,  2,    3;
%e A369287   0,  2,    4;
%e A369287   0,  4,   15,    8;
%e A369287   0,  4,   24,   19;
%e A369287   0,  7,   60,   79,    23;
%e A369287   0,  8,  101,  213,    84;
%e A369287   0, 12,  210,  615,   424,    66;
%e A369287   0, 14,  357, 1523,  1533,   363;
%e A369287   0, 21,  679, 3851,  5580,  2217,  212;
%e A369287   0, 24, 1142, 8963, 17836, 10379, 1575;
%e A369287   ...
%e A369287 The T(5,1) = 2 multiset partitions are:
%e A369287    {{1,1,1,1,1}},
%e A369287    {{1,1,1,2,2}}.
%e A369287 The corresponding T(5,1) = 2 matrices are:
%e A369287    [5]  [3 2].
%e A369287 The T(5,2) = 4 matrices are:
%e A369287    [3]  [3 0]  [2 1]  [2 1]
%e A369287    [2]  [0 2]  [1 1]  [0 2],
%o A369287 (PARI)
%o A369287 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A369287 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 A369287 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 A369287 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 A369287 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 A369287 T(n)={my(v=Vec(G(n,'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
%o A369287 { my(A=T(15)); for(i=1, #A, print(A[i])) }
%Y A369287 Row sums are A320665.
%Y A369287 Columns k=0..1 are A000007, A002865(n>0).
%Y A369287 Cf. A050535, A369286.
%K A369287 nonn,tabf
%O A369287 0,8
%A A369287 _Andrew Howroyd_, Jan 28 2024