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
Examples
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;
Links
- 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.
Crossrefs
Cf. A074605.
Programs
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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
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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
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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 *)
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).
Comments