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A117279
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Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.
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5
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1, 1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 3, 1, 10, 45, 110, 140, 60, 10, 1, 15, 105, 435, 1125, 1701, 1200, 480, 105, 10, 1, 21, 210, 1295, 5355, 14952, 26572, 26670, 17535, 7840, 2331, 420, 35, 1, 28, 378, 3220, 19075, 81228, 246414, 507424, 666015, 620900, 431368
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OFFSET
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0,6
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.
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LINKS
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FORMULA
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E.g.f.: sqrt(Sum_{n>=0} exp(x*(1+q)^n)*x^n/n!).
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EXAMPLE
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Triangle begins:
1;
1;
1, 1;
1, 3, 3;
1, 6, 15, 16, 3;
1, 10, 45, 110, 140, 60, 10;
...
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MATHEMATICA
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nn=10; f[x_, y_]:=Sum[Sum[Binomial[n, k](1+y)^(k(n-k)), {k, 0, n}]x^n/n!, {n, 0, nn}]; Map[Select[#, #>0&]&, Range[0, nn]!CoefficientList[Series[Exp[Log[f[x, y]]/2], {x, 0, nn}], {x, y}]]//Grid (* Geoffrey Critzer, Sep 05 2013 *)
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PROG
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(PARI)
T(n)={[Vecrev(p) | p<-Vec(serlaplace(sqrt(sum(k=0, n, exp(x*(1+y)^k + O(x*x^n))*x^k/k! ))))]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 10 2022
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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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