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A144228
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Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph has at most one cycle.
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5
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1, 1, 0, 1, 1, 0, 1, 3, 3, 1, 1, 6, 15, 20, 15, 1, 10, 45, 120, 210, 222, 1, 15, 105, 455, 1365, 2913, 3670, 1, 21, 210, 1330, 5985, 20139, 49294, 68820, 1, 28, 378, 3276, 20475, 97860, 362670, 976560, 1456875, 1, 36, 630, 7140, 58905, 376236, 1914276, 7663500, 22089870, 34506640
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OFFSET
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0,8
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LINKS
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FORMULA
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T(n,0) = 1, T(n,k) = 0 if k<0 or n<k, else T(n,k) = Sum_{j=0..k} C(n-1,j) (A000272(j+1) T(n-j-1,k-j) + A057500(j+1) T(n-j-1,k-j-1)).
E.g.f.: exp(B(x,y)), where B(x,y) = Sum(Sum(A062734(n,k)*y^k*x^n/n!, k=0..n), n=1..infinity) = -1/2*log(1+LambertW(-x*y))+1/2*LambertW(-x*y) -1/4*LambertW(-x*y)^2-1/y *(LambertW(-x*y)+1/2 *LambertW(-x*y)^2). - Vladeta Jovovic, Sep 16 2008
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EXAMPLE
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T(4,4) = 15, because there are 15 simple graphs on 4 labeled nodes with 4 edges where each maximally connected subgraph has at most one cycle:
1-2 1-2 1-2 1-2 1-2 1-2 1 2 1 2 1-2 1 2 1 2 1-2 1-2 1-2 1 2
|/| |X |/ |\| X| \| |/| X| /| |\| |X |\ | | X |X|
4 3 4 3 4-3 4 3 4 3 4-3 4-3 4-3 4-3 4-3 4-3 4-3 4-3 4-3 4 3
Triangle begins:
1;
1, 0;
1, 1, 0;
1, 3, 3, 1;
1, 6, 15, 20, 15;
1, 10, 45, 120, 210, 222;
...
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MAPLE
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cy:= proc(n) option remember; local t; binomial(n-1, 2) *add((n-3)! /(n-2-t)! *n^(n-2-t), t=1..n-2) end: T:= proc(n, k) option remember; local j; if k=0 then 1 elif k<0 or n<k then 0 else add(binomial(n-1, j) *((j+1)^(j-1) *T(n-j-1, k-j) +cy(j+1) *T(n-j-1, k-j-1)), j=0..k) fi end: seq(seq(T(n, k), k=0..n), n=0..11);
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MATHEMATICA
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t[_, 0] = 1; t[n_, k_] /; (k<0 || n<k) = 0; t[n_, k_] := t[n, k] = Sum[Binomial[n-1, j]*(t[n-j-1, k-j]*(j+1)^(j-1) + 1/2*j!*Sum[1/((j+1)^m*(j-m+1)!), {m, 3, j+1}]*t[n-j-1, k-j-1]*(j+1)^(j+1)), {j, 0, k}]; Flatten[Table[t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Jan 15 2014, after Maple *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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