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A123551
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Triangle read by rows: T(n,k) gives number of unlabeled graphs without endpoints on n nodes and k edges, (n >= 0, 0 <= k <= n(n-1)/2).
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5
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1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 4, 3, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 10, 22, 48, 76, 97, 102, 84, 60, 39, 20, 10, 5, 2, 1, 1, 1, 0, 0, 1, 1, 2, 6, 10, 25, 64, 152, 331, 617, 930, 1173, 1253, 1140
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OFFSET
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0,21
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REFERENCES
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R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
[0] 1;
[1] 1;
[2] 1, 0;
[3] 1, 0, 0, 1;
[4] 1, 0, 0, 1, 1, 1, 1;
[5] 1, 0, 0, 1, 1, 2, 4, 3, 2, 1, 1;
[6] 1, 0, 0, 1, 1, 2, 6, 8, 13, 16, 13, 8, 5, 2, 1, 1;
...
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PROG
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(PARI)
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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
row(n) = {my(s=0); sum(k=0, n, forpart(p=k, s+=permcount(p) * edges(p, w->1+y^w) * y^(n-k)*polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); Vecrev(s, binomial(n, 2)+1)}
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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