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A318602
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Triangle read by rows: T(n,k) is the number of rooted hypertrees on n unlabeled nodes with k edges, (0 <= k < n).
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3
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1, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 5, 10, 9, 0, 1, 6, 20, 30, 20, 0, 1, 8, 33, 77, 91, 48, 0, 1, 9, 49, 152, 277, 268, 115, 0, 1, 11, 68, 269, 655, 969, 790, 286, 0, 1, 12, 91, 428, 1330, 2651, 3294, 2308, 719, 0, 1, 14, 116, 647, 2420, 6137, 10300, 10993, 6737, 1842
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OFFSET
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1,6
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COMMENTS
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Equivalently, the number of rooted connected graphs on n unlabeled nodes with k blocks where every block is a complete graph.
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LINKS
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 1, 2;
0, 1, 3, 4;
0, 1, 5, 10, 9;
0, 1, 6, 20, 30, 20;
0, 1, 8, 33, 77, 91, 48;
0, 1, 9, 49, 152, 277, 268, 115;
0, 1, 11, 68, 269, 655, 969, 790, 286;
...
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PROG
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(PARI)
EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)}
R(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- v]}
{ my(T=R(10)); for(n=1, #T, print(T[n])) }
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CROSSREFS
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Rightmost diagonal is A000081 (rooted trees).
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KEYWORD
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AUTHOR
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STATUS
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approved
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