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A330693
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Regular subtriangle of A328773: T(n,k) is the number of the balanced colored digraphs on n nodes with k colors, 1 <= k <= n.
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0
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1, 3, 4, 16, 36, 64, 218, 1104, 2112, 4096, 9608, 90416, 266496, 528384, 1048576, 1540944, 30194176, 135032832, 269500416, 537919488, 1073741824
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..21.
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FORMULA
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T(n,1) = A000273(n).
T(n,n) = A053763(n) = 2^(n^2 - n).
T(n,k) = A328773(n,A330661(n,k)).
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EXAMPLE
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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.
Triangle T(n,k) begins:
1;
3, 4;
16, 36, 64;
218, 1104, 2112, 4096;
9608, 90416, 266496, 528384, 1048576;
1540944, 30194176, 135032832, 269500416, 537919488, 1073741824;
...
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PROG
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(PARI)
\\ here C(p) computes sequence value for given partition.
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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}
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, [])}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n), p[#p]-p[1]<=1], , 4))}
{ for(n=1, 6, print(Row(n))) }
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CROSSREFS
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Cf. A328773, A330661, A000273, A053763.
Sequence in context: A290433 A251582 A328773 * A329541 A154736 A188114
Adjacent sequences: A330690 A330691 A330692 * A330694 A330695 A330696
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KEYWORD
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nonn,tabl,more
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AUTHOR
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Peter Dolland, Dec 25 2019
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STATUS
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approved
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