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A320801 Regular triangle read by rows where T(n,k) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n.

%I A320801 #11 Jan 16 2024 17:32:51
%S A320801 1,0,1,0,1,3,0,1,3,6,0,1,6,10,16,0,1,6,20,30,34,0,1,9,31,75,92,90,0,1,
%T A320801 9,45,126,246,272,211,0,1,12,60,223,501,839,823,558,0,1,12,81,324,953,
%U A320801 1900,2762,2482,1430,0,1,15,100,491,1611,4033,7120,9299,7629,3908
%N A320801 Regular triangle read by rows where T(n,k) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n.
%H A320801 Andrew Howroyd, <a href="/A320801/b320801.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A320801 Triangle begins:
%e A320801    1
%e A320801    0   1
%e A320801    0   1   3
%e A320801    0   1   3   6
%e A320801    0   1   6  10  16
%e A320801    0   1   6  20  30  34
%e A320801    0   1   9  31  75  92  90
%e A320801    0   1   9  45 126 246 272 211
%e A320801    0   1  12  60 223 501 839 823 558
%o A320801 (PARI)
%o A320801 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 A320801 K(q, t, k)={prod(j=1, #q, my(g=gcd(t, q[j]), e=(q[j]/g)); (1 - y^e + y^e/(1 - x^e) + O(x*x^k))^g) - 1}
%o A320801 G(n)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, substvec(K(q, t, n\t)/t, [x,y], [x^t,y^t])) + O(x*x^n))); s/n!}
%o A320801 T(n)=[Vecrev(p) | p<-Vec(G(n))]
%o A320801 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 16 2024
%Y A320801 Row sums are A007716. Last column is A049311.
%Y A320801 Cf. A316980, A316983, A317533, A318805, A319616, A319721, A320796, A320808, A321194.
%K A320801 nonn,tabl
%O A320801 0,6
%A A320801 _Gus Wiseman_, Nov 09 2018
%E A320801 Offset corrected by _Andrew Howroyd_, Jan 16 2024