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A106239
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Triangle read by rows: T(n,m) = number of graphs on n labeled nodes, with m components of size = order. Also number of graphs on n labeled nodes with m unicyclic components.
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4
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0, 0, 0, 1, 0, 0, 15, 0, 0, 0, 222, 0, 0, 0, 0, 3660, 10, 0, 0, 0, 0, 68295, 525, 0, 0, 0, 0, 0, 1436568, 20307, 0, 0, 0, 0, 0, 0, 33779340, 727020, 280, 0, 0, 0, 0, 0, 0, 880107840, 25934184, 31500, 0, 0, 0, 0, 0, 0, 0, 25201854045, 950478210, 2325015, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,7
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LINKS
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Alois P. Heinz, Rows n = 1..141, flattened
Washington Bomfim, Illustration of this sequence
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FORMULA
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E.g.f.: exp(-y/2*ln(1+LambertW(-x))+y/2*LambertW(-x)-y/4*LambertW(-x)^2). - Vladeta Jovovic, May 04 2005
T(n,m) = sum N/D over the partitions of n:1K1+2K2+ ... +nKn, = with exactly m parts greater than 2, where N = n!*product_{1=<i<=n}= A057500(i)^Ki and D = product_{1=<i<=n}(Ki!(i!)^Ki).
T(n,1) = A057500(n), T(n,m) = Sum_{j=2..n-1} C(n-1,j) * A057500(j+1) * T(n-1-j,m-1) if m>1. - Alois P. Heinz, Sep 15 2008
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EXAMPLE
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T(6,2) = 10 because there are 10 such graphs of order 6 with 2 components. The value of T(n,m) is zero if and only if m > floor(n/3).
Triangle T(n,m) begins:
0;
0, 0;
1, 0, 0;
15, 0, 0, 0;
222, 0, 0, 0, 0;
3660, 10, 0, 0, 0, 0;
68295, 525, 0, 0, 0, 0, 0;
1436568, 20307, 0, 0, 0, 0, 0, 0;
33779340, 727020, 280, 0, 0, 0, 0, 0, 0;
880107840, 25934184, 31500, 0, 0, 0, 0, 0, 0, 0;
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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, m) if m=1 then cy(n) else add (binomial(n-1, j) *cy(j+1) * T(n-1-j, m-1), j=2..n-1) fi end: seq (seq (T(n, m), m=1..n), n=1..11); # Alois P. Heinz, Sep 15 2008
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MATHEMATICA
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nn=12; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[ Series[Exp[y(Log[1/(1-t)]/2-t/2-t^2/4)], {x, 0, nn}], {x, y}] //Grid (* Geoffrey Critzer, Nov 04 2012 *)
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CROSSREFS
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Cf. A057500 and A106238 (similar formulae that can be used in the unlabeled case).
Sequence in context: A123652 A127622 A185294 * A202857 A055965 A067154
Adjacent sequences: A106236 A106237 A106238 * A106240 A106241 A106242
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KEYWORD
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nonn,tabl
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AUTHOR
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Washington Bomfim, May 03 2005
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STATUS
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approved
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