A378418 Irregular triangle read by rows: T(n,k) is the coefficient of x^k in the domination polynomial of the n X n torus grid graph (n>=1, A094087(n)<=k<=n^2).

1, 6, 4, 1, 48, 117, 126, 84, 36, 9, 1, 40, 560, 2736, 6800, 10310, 10560, 7832, 4352, 1820, 560, 120, 16, 1, 10, 200, 3050, 31525, 188700, 677690, 1610700, 2740775, 3527075, 3562700, 2895610, 1923600, 1053175, 475950, 176600, 53105, 12650, 2300, 300, 25, 1, 18

Offset

1,2

Comments

Extended to n=1.

Sum_{k=A094087(n)..n^2} T(n,k) = A303334(n).

T(n,n^2) = 1.

Links

Eric W. Weisstein, Table of n, a(n) for n = 1..175

Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], Aug 2024.

Eric Weisstein's World of Mathematics, Dominating Set.

Eric Weisstein's World of Mathematics, Domination Number.

Eric Weisstein's World of Mathematics, Domination Polynomial.

Eric Weisstein's World of Mathematics, Torus Grid Graph.

Example

D(1) = x
D(2)= 6*x^2+4*x^3+x^4
D(3) = 48*x^3+117*x^4+126*x^5+84*x^6+36*x^7+9*x^8+x^9
D(4) = 40*x^4+560*x^5+2736*x^6+6800*x^7+10310*x^8+10560*x^9+7832*x^10+4352*x^11+1820*x^12+560*x^13+120*x^14+16*x^15+x^16

Crossrefs

Cf. A094087 (domination number of the n X n torus grid graph).

Cf. A303334 (number of dominating sets in the n X n torus grid graph).

Cf. A000290 (vertex count of the n X n torus grid graph = n^2).

Sequence in context: A378412 A204023 A360984 * A166905 A278071 A362191

Adjacent sequences: A378415 A378416 A378417 * A378419 A378420 A378421

Keyword

nonn,tabf,hard

Author

Eric W. Weisstein, Nov 25 2024

Status

approved