A212633 Triangle read by rows: T(n,k) is the number of dominating subsets with cardinality k of the path tree P_n (n>=1, 1<=k<=n).

1, 2, 1, 1, 3, 1, 0, 4, 4, 1, 0, 3, 8, 5, 1, 0, 1, 10, 13, 6, 1, 0, 0, 8, 22, 19, 7, 1, 0, 0, 4, 26, 40, 26, 8, 1, 0, 0, 1, 22, 61, 65, 34, 9, 1, 0, 0, 0, 13, 70, 120, 98, 43, 10, 1, 0, 0, 0, 5, 61, 171, 211, 140, 53, 11, 1, 0, 0, 0, 1, 40, 192, 356, 343, 192, 64, 12, 1

Offset

1,2

Comments

The entries in row n are the coefficients of the domination polynomial of the path P_n (see the Alikhani and Peng reference).

Sum of entries in row n = A000213(n+1) (number of dominating subsets; tribonacci numbers).

References

S. Alikhani and Y. H. Peng, Dominating sets and domination polynomials of paths, International J. Math. and Math. Sci., Vol. 2009, Article ID542040.

Links

Alois P. Heinz, Rows n = 1..141, flattened

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251.

JL Arocha, B Llano, The number of dominating k-sets of paths, cycles and wheels, arXiv preprint arXiv:1601.01268, 2016

Eric Weisstein's World of Mathematics, Domination Polynomial

Eric Weisstein's World of Mathematics, Path Graph

Formula

If p(n)=p(n,x) denotes the generating polynomial of row n (called the domination polynomial of the path tree P_n), then p(1)=x, p(2) = 2x + x^2, p(3) = x + 3x^2 + x^3 and p(n) = x*[p(n-1) + p(n-2) + p(n-3)] for n>=4 (see Eq. (3.2) in the Alikhani & Peng journal reference).

Example

Row 3 is [1,3,1] because the path tree A-B-C has dominating subsets B, AB, BC, AC, and ABC.
Triangle starts:
1;
2,1;
1,3,1;
0,4,4,1;
0,3,8,5,1;

Maple

p := proc (n) option remember; if n = 1 then x elif n = 2 then x^2+2*x elif n = 3 then x^3+3*x^2+x else sort(expand(x*(p(n-1)+p(n-2)+p(n-3)))) end if end proc: for n to 15 do seq(coeff(p(n), x, k), k = 1 .. n) end do; # yields sequence in triangular form

Mathematica

p[1] = x; p[2] = 2*x + x^2; p[3] = x + 3*x^2 + x^3; p[n_] := p[n] = x*(p[n - 1] + p[n - 2] + p[n - 3]); row[n_] := CoefficientList[p[n], x] // Rest;
Array[row, 15] // Flatten (* Jean-François Alcover, Feb 24 2016 *)
CoefficientList[LinearRecurrence[{x, x, x}, {1, 2 + x, 1 + 3 x + x^2}, 10], x] // Flatten (* Eric W. Weisstein, Apr 07 2017 *)

Crossrefs

Cf. A000213, A212634, A212635.

Sequence in context: A146014 A344824 A226173 * A202241 A248156 A352780

Adjacent sequences: A212630 A212631 A212632 * A212634 A212635 A212636

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 14 2012

Status

approved