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A249632
Triangular array read by rows. T(n,k) is the number of labeled trees with black and white nodes having exactly k black nodes, n>=0, 0<=k<=n.
0
1, 1, 1, 1, 2, 1, 3, 9, 9, 3, 16, 64, 96, 64, 16, 125, 625, 1250, 1250, 625, 125, 1296, 7776, 19440, 25920, 19440, 7776, 1296, 16807, 117649, 352947, 588245, 588245, 352947, 117649, 16807, 262144, 2097152, 7340032, 14680064, 18350080, 14680064, 7340032, 2097152, 262144
OFFSET
0,5
COMMENTS
Row sums = A038058.
T(n,n) = T(n,0) = n^(n-2) free trees A000272.
T(n,n-1) = T(n,1) = n^(n-1) rooted trees A000169.
T(n,2) = A081131.
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press,1973, page 30, exercise 1.10.
FORMULA
E.g.f.: A(x + y*x) where A(x) is the e.g.f. for A000272.
EXAMPLE
1,
1, 1,
1, 2, 1,
3, 9, 9, 3,
16, 64, 96, 64, 16,
125, 625, 1250, 1250, 625, 125,
1296, 7776, 19440, 25920, 19440, 7776, 1296
MATHEMATICA
nn = 6; f[x_] := Sum[n^(n - 2) x^n/n!, {n, 1, nn}];
Map[Select[#, # > 0 &] &,
Range[0, nn]! CoefficientList[
Series[f[x + y x] + 1, {x, 0, nn}], {x, y}]] // Grid
CROSSREFS
Sequence in context: A234746 A359648 A237765 * A126009 A301282 A246063
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Nov 02 2014
STATUS
approved