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A241669 Irregular triangular array read by rows: T(n,k) is the number of 2-colored simple labeled graphs on n nodes that have exactly k edges, 0<=k<=A002620(n), n>=1.

%I A241669 #25 May 12 2025 07:07:48
%S A241669 0,2,2,6,12,6,14,48,60,32,6,30,160,360,440,310,120,20,62,480,1680,
%T A241669 3480,4680,4212,2520,960,210,20,126,1344,6720,20720,43680,66108,73514,
%U A241669 60480,36540,15820,4662,840,70,254,3584,24192,103040,308560,686784,1172976,1565888,1649340,1373680,900592,459312,178416,50960,10080,1232,70,510,9216,80640,451584,1808352,5491584,13102992,25128720,39312018,50638224,53981928,47698560,34869744,20975472,10281672,4044096,1246644,290304,48048,5040,252
%N A241669 Irregular triangular array read by rows: T(n,k) is the number of 2-colored simple labeled graphs on n nodes that have exactly k edges, 0<=k<=A002620(n), n>=1.
%H A241669 R. C. Read, <a href="https://doi.org/10.4153/CJM-1960-035-0">The number of k-colored graphs on labelled nodes</a>, Canad. J. Math., 12 (1960), 410-414.
%F A241669 E.g.f.: Sum_{n>=1} (exp(1 + y)^n*x - 1)*x^n/n!.
%e A241669 Triangle begins:
%e A241669   0,
%e A241669   2,  2,
%e A241669   6,  12,  6,
%e A241669   14, 48,  60,   32,   6,
%e A241669   30, 160, 360,  440,  310,  120,  20,
%e A241669   62, 480, 1680, 3480, 4680, 4212, 2520, 960, 210, 20
%t A241669 nn=10;f[x_]:=Sum[x^n/(n!*(1+y)^(n^2/2)),{n,0,nn}];CoefficientList[Table[n!*(1+y)^(n^2/2),{n,0,nn}]CoefficientList[Series[(f[x]-1)^2,{x,0,nn}],x]//Simplify//Expand,y]//Grid
%Y A241669 Cf. A002620, A213441 (row sums).
%K A241669 nonn,tabf
%O A241669 1,2
%A A241669 _Geoffrey Critzer_, Aug 08 2014