|
%I #21 Apr 18 2021 20:37:25
%S 1,1,3,6,3,7,24,30,16,3,15,80,180,220,155,60,10,31,240,840,1740,2340,
%T 2106,1260,480,105,10,63,672,3360,10360,21840,33054,36757,30240,18270,
%U 7910,2331,420,35,127,1792,12096,51520,154280,343392,586488,782944,824670,686840,450296,229656,89208,25480,5040,616,35
%N 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 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 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, page 16.
%H Andrew Howroyd, <a href="/A201143/b201143.txt">Table of n, a(n) for n = 2..1403</a> (rows 2..25)
%F O.g.f. of row n: Sum_{k=0..n-1} binomial(n,k)*(1+x)^(k*(n-k))/2.
%e Triangle begins:
%e 1, 1;
%e 3, 6, 3;
%e 7, 24, 30, 16, 3;
%e 15, 80, 180, 220, 155, 60, 10;
%e 31, 240, 840, 1740, 2340, 2106, 1260, 480, 105, 10;
%t Flatten[CoefficientList[Expand[Table[Sum[Binomial[n, k] (1 + x)^(k (n - k)), {k, 1, n - 1}]/2!, {n, 1,7}]], x]]
%o (PARI) Row(n) = {Vecrev(sum(k=1, n-1, binomial(n,k)*(1+x)^(k*(n-k))/2))}
%o { for(n=2, 8, print(Row(n))) } \\ _Andrew Howroyd_, Apr 18 2021
%Y Row sums are A058872.
%Y Row lengths appear to be A033638(n).
%K nonn,tabf
%O 2,3
%A _Geoffrey Critzer_, Nov 27 2011
%E Terms a(42) and beyond from _Andrew Howroyd_, Apr 18 2021
|
