1,2

Row n contains 2n + 2 entries.

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

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.

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).

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;

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

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 *)

(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

Cf. A213659.

Sequence in context: A124602 A320060 A132707 * A046575 A154739 A321044

Adjacent sequences: A213655 A213656 A213657 * A213659 A213660 A213661

nonn,tabf

Emeric Deutsch, Jun 29 2012

approved