A213660 Irregular triangle read by rows: T(n,k) is the number of dominating subsets with k vertices of the graph G(n) obtained by taking n copies of the cycle graph C_3 with a vertex in common.

3, 3, 1, 1, 8, 10, 5, 1, 1, 6, 23, 32, 21, 7, 1, 1, 8, 28, 72, 102, 80, 36, 9, 1, 1, 10, 45, 120, 242, 332, 290, 160, 55, 11, 1, 1, 12, 66, 220, 495, 856, 1116, 1032, 655, 280, 78, 13, 1, 1, 14, 91, 364, 1001, 2002, 3131, 3880, 3675, 2562, 1281, 448, 105, 15, 1

Offset

1,1

Comments

Row n contain 2n + 1 entries.

Sum of entries in row n = 3^n + 4^n = A074605(n).

Links

Table of n, a(n) for n=1..63.

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

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

Example

Row 1 is 3,3,1 because the graph G(1) is the triangle abc; there are 3 dominating subsets of size 1 ({a}, {b}, {c}), 3 dominating subsets of size 2 ({a,b}, {a,c}, {b,c}), and 1 dominating subset of size 3 ({a,b,c}).
T(n,1)=1 for n >= 2 because the common vertex of the triangles is the only dominating subset of size k=1.
Triangle starts:
  3, 3,  1;
  1, 8, 10,  5,   1;
  1, 6, 23, 32,  21,  7,  1;
  1, 8, 28, 72, 102, 80, 36, 9, 1;

Maple

T := proc (n, k) options operator, arrow: 2^(2*n-k)*binomial(n, k-n)+binomial(2*n, k-1) end proc: for n to 9 do seq(T(n, k), k = 1 .. 2*n+1) end do; # yields sequence in triangular form

Mathematica

T[n_, k_] := 2^(2n-k) Binomial[n, k-n] + Binomial[2n, k-1];
Table[T[n, k], {n, 1, 9}, {k, 1, 2n+1}] // Flatten (* Jean-François Alcover, Dec 06 2017 *)

Prog

(Magma) /* As triangle */ [[2^(2*n-k)*Binomial(n, k-n)+Binomial(2*n, k-1): k in [1..2*n+1]]: n in [1.. 10]]; // Vincenzo Librandi, Jul 20 2019

Crossrefs

Cf. A074605.

Sequence in context: A165795 A287290 A287981 * A099037 A340934 A271706

Adjacent sequences: A213657 A213658 A213659 * A213661 A213662 A213663

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 29 2012

Status

approved