A213658 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) consisting of an edge ab and n triangles, each having one vertex identified with the vertex b.

1, 5, 4, 1, 1, 5, 14, 14, 6, 1, 1, 7, 21, 43, 47, 27, 8, 1, 1, 9, 36, 84, 142, 158, 108, 44, 10, 1, 1, 11, 55, 165, 330, 494, 542, 410, 205, 65, 12, 1, 1, 13, 78, 286, 715, 1287, 1780, 1908, 1527, 875, 346, 90, 14, 1, 1, 15, 105, 455, 1365, 3003, 5005

Offset

1,2

Comments

Row n contains 2n + 2 entries.

Sum of entries in row n = 3^n + 2^{2n+1} = A213659(n).

Links

Table of n, a(n) for n=1..61.

S. Alikhani and E. Deutsch, Graphs with domination roots in the right half-plane, arXiv preprint arXiv:1305.3734 [math.CO], 2013-2014.

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251 [math.CO], 2009.

T. Kotek, J. Preen, F. Simon, P. Tittmann, and M. Trinks, Recurrence relations and splitting formulas for the domination polynomial, arXiv:1206.5926 [math.CO], 2012.

Formula

Generating polynomial of row n is x^(n+1)*(2+x)^n + x*(1+x)^(2*n+1); this is the domination polynomial of the graph G(n).

T(n,k) = 2^(2*n+1-k)*binomial(n,k-n-1) + binomial(2*n+1,k-1) (n >= 1; 1 <= k <= 2*n+2).

Example

Row 1 is 1,5,4,1 because the graph G(1) is abcd with edges ab, bc, bd, and cd; there is 1 dominating subset of size 1 ({b}); all binomial(4,2)=6 subsets of size 2 of {a,b,c,d} with the exception of {c,d} are dominating; all binomial(4,3)=4 subsets of size 3 of {a,b,c,d} are dominating; obviously, {a,b,c,d} is dominating.
Triangle starts:
  1,  5,  4,  1;
  1,  5, 14, 14,  6,  1;
  1,  7, 21, 43, 47, 27,  8,  1;

Maple

T := proc (n, k) options operator, arrow: 2^(2*n+1-k)*binomial(n, k-n-1)+binomial(2*n+1, k-1) end proc: for n to 8 do seq(T(n, k), k = 1 .. 2*n+2) end do; # yields sequence in triangular form

Mathematica

row[n_] := CoefficientList[x^(n + 1)*(2 + x)^n + x*(1 + x)^(2*n + 1), x] // Rest;
Table[row[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Dec 02 2017 *)

Prog

(Magma) /* As triangle */ [[2^(2*n+1-k)*Binomial(n, k-n-1) + Binomial(2*n+1, k-1): k in [1..2*n+2]]: n in [1.. 15]]; // Vincenzo Librandi, Jul 21 2019

Crossrefs

Cf. A213659.

Sequence in context: A124602 A320060 A132707 * A046575 A154739 A321044

Adjacent sequences: A213655 A213656 A213657 * A213659 A213660 A213661

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 29 2012

Status

approved