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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.

%I A249632 #12 Apr 26 2016 12:37:41
%S A249632 1,1,1,1,2,1,3,9,9,3,16,64,96,64,16,125,625,1250,1250,625,125,1296,
%T A249632 7776,19440,25920,19440,7776,1296,16807,117649,352947,588245,588245,
%U A249632 352947,117649,16807,262144,2097152,7340032,14680064,18350080,14680064,7340032,2097152,262144
%N 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.
%C A249632 Row sums = A038058.
%C A249632 T(n,n) = T(n,0) = n^(n-2) free trees A000272.
%C A249632 T(n,n-1) = T(n,1) = n^(n-1) rooted trees A000169.
%C A249632 T(n,2) = A081131.
%D A249632 F. Harary and E. Palmer, Graphical Enumeration, Academic Press,1973, page 30, exercise 1.10.
%F A249632 E.g.f.: A(x + y*x) where A(x) is the e.g.f. for A000272.
%e A249632 1,
%e A249632 1,    1,
%e A249632 1,    2,    1,
%e A249632 3,    9,    9,     3,
%e A249632 16,   64,   96,    64,    16,
%e A249632 125,  625,  1250,  1250,  625,   125,
%e A249632 1296, 7776, 19440, 25920, 19440, 7776, 1296
%t A249632 nn = 6; f[x_] := Sum[n^(n - 2) x^n/n!, {n, 1, nn}];
%t A249632 Map[Select[#, # > 0 &] &,
%t A249632   Range[0, nn]! CoefficientList[
%t A249632     Series[f[x + y x] + 1, {x, 0, nn}], {x, y}]] // Grid
%K A249632 nonn,tabl
%O A249632 0,5
%A A249632 _Geoffrey Critzer_, Nov 02 2014