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A343088
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Triangle read by rows: T(n,k) is the number of connected labeled graphs with n edges and k vertices, 1 <= k <= n+1.
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14
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1, 0, 1, 0, 0, 3, 0, 0, 1, 16, 0, 0, 0, 15, 125, 0, 0, 0, 6, 222, 1296, 0, 0, 0, 1, 205, 3660, 16807, 0, 0, 0, 0, 120, 5700, 68295, 262144, 0, 0, 0, 0, 45, 6165, 156555, 1436568, 4782969, 0, 0, 0, 0, 10, 4945, 258125, 4483360, 33779340, 100000000
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listen;
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text;
internal format)
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OFFSET
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0,6
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LINKS
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 0, 3;
0, 0, 1, 16;
0, 0, 0, 15, 125;
0, 0, 0, 6, 222, 1296;
0, 0, 0, 1, 205, 3660, 16807;
0, 0, 0, 0, 120, 5700, 68295, 262144;
0, 0, 0, 0, 45, 6165, 156555, 1436568, 4782969;
...
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MATHEMATICA
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row[n_] := (SeriesCoefficient[#, {y, 0, n}]& /@ CoefficientList[ Log[Sum[x^k*(1+y)^Binomial[k, 2]/k!, {k, 0, n+1}]] + O[x]^(n+2), x]* Range[0, n+1]!) // Rest;
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PROG
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(PARI)
Row(n)={Vec(serlaplace(polcoef(log(O(x^2*x^n)+sum(k=0, n+1, x^k*(1 + y + O(y*y^n))^binomial(k, 2)/k!)), n, y)), -(n+1))}
{ for(n=0, 8, print(Row(n))) }
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CROSSREFS
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Subsequent diagonals give the number of connected labeled graphs with n nodes and n+k edges for k=0..11: A057500, A061540, A061541, A061542, A061543, A096117, A061544 A096150, A096224, A182294, A182295, A182371.
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KEYWORD
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AUTHOR
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STATUS
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approved
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