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A285529 Triangle read by rows: T(n,k) is the number of nodes of degree k counted over all simple labeled graphs on n nodes, n>=1, 0<=k<=n-1.

%I A285529 #14 Apr 21 2017 08:30:02
%S A285529 1,2,2,6,12,6,32,96,96,32,320,1280,1920,1280,320,6144,30720,61440,
%T A285529 61440,30720,6144,229376,1376256,3440640,4587520,3440640,1376256,
%U A285529 229376,16777216,117440512,352321536,587202560,587202560,352321536,117440512,16777216
%N A285529 Triangle read by rows: T(n,k) is the number of nodes of degree k counted over all simple labeled graphs on n nodes, n>=1, 0<=k<=n-1.
%F A285529 E.g.f. for column k: x * Sum_{n>=0} binomial(n,k)*2^binomial(n,2)*x^n/n!.
%F A285529 Sum_{k=1..n-1} T(n,k)*k/2 = A095351(n).
%F A285529 T(n,k) = n*binomial(n-1,k)*2^binomial(n-1,2). - _Alois P. Heinz_, Apr 21 2017
%e A285529 1,
%e A285529 2,   2,
%e A285529 6,   12,   6,
%e A285529 32,  96,   96,   32,
%e A285529 320, 1280, 1920, 1280, 320,
%e A285529 ...
%t A285529 nn = 9; Map[Select[#, # > 0 &] &,
%t A285529   Drop[Transpose[Table[A[z_] := Sum[Binomial[n, k] 2^Binomial[n, 2] z^n/n!, {n, 0, nn}];Range[0, nn]! CoefficientList[Series[z A[z], {z, 0, nn}], z], {k,0, nn - 1}]], 1]] // Grid
%Y A285529 Row sums give A095340.
%Y A285529 Columns for k=0-3: A123903, A095338, A038094, A038096.
%K A285529 nonn,tabl
%O A285529 1,2
%A A285529 _Geoffrey Critzer_, Apr 20 2017