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

2,3

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.

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)

Formula

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

Example

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;

Mathematica

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

Prog

(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

Crossrefs

Row sums are A058872.

Row lengths appear to be A033638(n).

Sequence in context: A170859 A123146 A016661 * A326935 A135003 A350877

Adjacent sequences: A201140 A201141 A201142 * A201144 A201145 A201146

Keyword

nonn,tabf

Author

Geoffrey Critzer, Nov 27 2011

Extensions

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

Status

approved