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A369928
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Triangle read by rows: T(n,k) is the number of simple graphs on n labeled vertices with k edges and without endpoints, n >= 0, 0 <= k <= n*(n-1)/2.
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3
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1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 4, 3, 6, 1, 1, 0, 0, 10, 15, 42, 90, 100, 45, 10, 1, 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1, 1, 0, 0, 35, 105, 462, 2310, 9495, 32130, 85365, 166341, 231861, 237125, 184380, 111870, 53634, 20307, 5985, 1330, 210, 21, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,12
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LINKS
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FORMULA
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E.g.f.: exp(y*x^2/2) * Sum_{k>=0} (1 + y)^binomial(k, 2)*(x/exp(y*x))^k/k!.
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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, 4, 3, 6, 1;
[5] 1, 0, 0, 10, 15, 42, 90, 100, 45, 10, 1;
[6] 1, 0, 0, 20, 45, 162, 595, 1590, 3075, 3655, 2703, 1335, 455, 105, 15, 1;
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PROG
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(PARI) \\ row(n) gives n-th row as vector.
row(n)={my(A=x/exp(x*y + O(x*x^n))); Vecrev(polcoef(serlaplace(exp(y*x^2/2 + O(x*x^n)) * sum(k=0, n, (1 + y)^binomial(k, 2)*A^k/k!)), n), 1 + binomial(n, 2))}
{ for(n=0, 6, print(row(n))) }
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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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