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A330693 Regular subtriangle of A328773: T(n,k) is the number of the balanced colored digraphs on n nodes with k colors, 1 <= k <= n.

%I A330693 #9 Feb 21 2020 10:15:37
%S A330693 1,3,4,16,36,64,218,1104,2112,4096,9608,90416,266496,528384,1048576,
%T A330693 1540944,30194176,135032832,269500416,537919488,1073741824
%N A330693 Regular subtriangle of A328773: T(n,k) is the number of the balanced colored digraphs on n nodes with k colors, 1 <= k <= n.
%C A330693 The color count, k, is a measure of the differentiation of the nodes: k = 1 means no differentiation at all, so we have A000273. k = n means all nodes are taken to be different, and we have A053763. With increasing differentiation T(n,k) < T(n,k+1) for 1 <= k < n.
%F A330693 T(n,1) = A000273(n).
%F A330693 T(n,n) = A053763(n) = 2^(n^2 - n).
%F A330693 T(n,k) = A328773(n,A330661(n,k)).
%e A330693 The balanced color scheme [2,2,2] has with A330661 the index 5 in A328773. So there are T(6,3) = A328773(6,5) = 135032832 digraphs with 6 nodes two each in the same color.
%e A330693 Triangle T(n,k) begins:
%e A330693         1;
%e A330693         3,        4;
%e A330693        16,       36,        64;
%e A330693       218,     1104,      2112,      4096;
%e A330693      9608,    90416,    266496,    528384,   1048576;
%e A330693   1540944, 30194176, 135032832, 269500416, 537919488, 1073741824;
%e A330693   ...
%o A330693 (PARI)
%o A330693 \\ here C(p) computes sequence value for given partition.
%o A330693 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 A330693 edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}
%o A330693 C(p)={((i, v)->if(i>#p, 2^edges(v), my(s=0); forpart(q=p[i], s+=permcount(q)*self()(i+1, concat(v, Vec(q)))); s/p[i]!))(1, [])}
%o A330693 Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n),p[#p]-p[1]<=1], , 4))}
%o A330693 { for(n=1, 6, print(Row(n))) }
%Y A330693 Cf. A328773, A330661, A000273, A053763.
%K A330693 nonn,tabl,more
%O A330693 1,2
%A A330693 _Peter Dolland_, Dec 25 2019