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A304942 Triangle read by rows: T(n,k) is the number of nonisomorphic binary n X n matrices with k 1's per column under row and column permutations.

%I A304942 #24 Aug 29 2018 20:42:35
%S A304942 1,1,1,1,2,1,1,3,3,1,1,5,11,5,1,1,7,35,35,7,1,1,11,132,410,132,11,1,1,
%T A304942 15,471,6178,6178,471,15,1,1,22,1806,122038,594203,122038,1806,22,1,1,
%U A304942 30,7042,2921607,85820809,85820809,2921607,7042,30,1
%N A304942 Triangle read by rows: T(n,k) is the number of nonisomorphic binary n X n matrices with k 1's per column under row and column permutations.
%H A304942 Andrew Howroyd, <a href="/A304942/b304942.txt">Table of n, a(n) for n = 0..527</a>
%H A304942 StackExchange, <a href="http://math.stackexchange.com/questions/616834/">How many arrays with crossed cells, order of rows/columns irrelevant</a>, Dec 13 2013
%e A304942 Triangle begins (n >=0, k >= 0):
%e A304942   1;
%e A304942   1,  1;
%e A304942   1,  2,    1;
%e A304942   1,  3,    3,      1;
%e A304942   1,  5,   11,      5,      1;
%e A304942   1,  7,   35,     35,      7,      1;
%e A304942   1, 11,  132,    410,    132,     11,    1;
%e A304942   1, 15,  471,   6178,   6178,    471,   15,  1;
%e A304942   1, 22, 1806, 122038, 594203, 122038, 1806, 22, 1;
%e A304942   ...
%o A304942 (PARI)
%o A304942 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 A304942 K(q,t,k)={polcoeff(prod(j=1, #q, my(g=gcd(t, q[j])); (1 + x^(q[j]/g) + O(x*x^k))^g), k)}
%o A304942 Blocks(n,m,k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoeff(exp(sum(t=1, n, K(q,t,k)/t*x^t) + O(x*x^n)), n)); s/m!}
%o A304942 for(n=0, 10, for(k=0, n, print1(Blocks(n,n,k), ", ")); print)
%Y A304942 Columns k=1..5 are A000041, A247417, A247596, A247597, A247598.
%Y A304942 Cf. A305027.
%K A304942 nonn,tabl
%O A304942 0,5
%A A304942 _Andrew Howroyd_, May 23 2018