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).

1, 1, 3, 6, 3, 7, 24, 30, 16, 3, 15, 80, 180, 220, 155, 60, 10, 31, 240, 840, 1740, 2340, 2106, 1260, 480, 105, 10, 63, 672, 3360, 10360, 21840, 33054, 36757, 30240, 18270, 7910, 2331, 420, 35, 127, 1792, 12096, 51520, 154280, 343392, 586488, 782944, 824670, 686840, 450296, 229656, 89208, 25480, 5040, 616, 35

Offset: 2

Author

Geoffrey Critzer, Nov 27 2011

Keywords

• nonn
• tabf

Comments

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.

Examples

			Triangle begins:
   1,   1;
   3,   6,   3;
   7,  24,  30,   16,    3;
  15,  80, 180,  220,  155,   60,   10;
  31, 240, 840, 1740, 2340, 2106, 1260, 480, 105, 10;
		

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, page 16.

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..1403 (rows 2..25)

Crossrefs

Row sums are A058872.

Row lengths appear to be A033638(n).

Programs

Mathematica

Flatten[CoefficientList[Expand[Table[Sum[Binomial[n, k] (1 + x)^(k (n - k)), {k, 1, n - 1}]/2!, {n, 1,7}]], x]]

PARI

Row(n) = {Vecrev(sum(k=1, n-1, binomial(n,k)*(1+x)^(k*(n-k))/2))}
{ for(n=2, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 18 2021

Formula

O.g.f. of row n: Sum_{k=0..n-1} binomial(n,k)*(1+x)^(k*(n-k))/2.

Extensions

Terms a(42) and beyond from Andrew Howroyd, Apr 18 2021