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%I #11 Jan 16 2024 17:32:51
%S 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 9,45,126,246,272,211,0,1,12,60,223,501,839,823,558,0,1,12,81,324,953,
%U 1900,2762,2482,1430,0,1,15,100,491,1611,4033,7120,9299,7629,3908
%N 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 Andrew Howroyd, <a href="/A320801/b320801.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 3
%e 0 1 3 6
%e 0 1 6 10 16
%e 0 1 6 20 30 34
%e 0 1 9 31 75 92 90
%e 0 1 9 45 126 246 272 211
%e 0 1 12 60 223 501 839 823 558
%o (PARI)
%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)={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 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 T(n)=[Vecrev(p) | p<-Vec(G(n))]
%o { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 16 2024
%Y Row sums are A007716. Last column is A049311.
%Y Cf. A316980, A316983, A317533, A318805, A319616, A319721, A320796, A320808, A321194.
%K nonn,tabl
%O 0,6
%A _Gus Wiseman_, Nov 09 2018
%E Offset corrected by _Andrew Howroyd_, Jan 16 2024
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