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A201143 Irregular triangular array read by rows T(n,k) is the number of 2-colored labeled graphs that have exactly k edges, n >= 2, 0 <= k <= A033638(n).

%I A201143 #21 Apr 18 2021 20:37:25
%S A201143 1,1,3,6,3,7,24,30,16,3,15,80,180,220,155,60,10,31,240,840,1740,2340,
%T A201143 2106,1260,480,105,10,63,672,3360,10360,21840,33054,36757,30240,18270,
%U A201143 7910,2331,420,35,127,1792,12096,51520,154280,343392,586488,782944,824670,686840,450296,229656,89208,25480,5040,616,35
%N A201143 Irregular triangular array read by rows T(n,k) is the number of 2-colored labeled graphs that have exactly k edges, n >= 2, 0 <= k <= A033638(n).
%C A201143 In each such graph: (i) no two nodes of the same color are adjacent,  (ii) the colors are interchangeable, and (iii) there must be at least one vertex of each color.
%D A201143 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, page 16.
%H A201143 Andrew Howroyd, <a href="/A201143/b201143.txt">Table of n, a(n) for n = 2..1403</a> (rows 2..25)
%F A201143 O.g.f. of row n: Sum_{k=0..n-1} binomial(n,k)*(1+x)^(k*(n-k))/2.
%e A201143 Triangle begins:
%e A201143    1,   1;
%e A201143    3,   6,   3;
%e A201143    7,  24,  30,   16,    3;
%e A201143   15,  80, 180,  220,  155,   60,   10;
%e A201143   31, 240, 840, 1740, 2340, 2106, 1260, 480, 105, 10;
%t A201143 Flatten[CoefficientList[Expand[Table[Sum[Binomial[n, k] (1 + x)^(k (n - k)), {k, 1, n - 1}]/2!, {n, 1,7}]], x]]
%o A201143 (PARI) Row(n) = {Vecrev(sum(k=1, n-1, binomial(n,k)*(1+x)^(k*(n-k))/2))}
%o A201143 { for(n=2, 8, print(Row(n))) } \\ _Andrew Howroyd_, Apr 18 2021
%Y A201143 Row sums are A058872.
%Y A201143 Row lengths appear to be A033638(n).
%K A201143 nonn,tabf
%O A201143 2,3
%A A201143 _Geoffrey Critzer_, Nov 27 2011
%E A201143 Terms a(42) and beyond from _Andrew Howroyd_, Apr 18 2021