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

Author

Eric W. Weisstein, Nov 25 2024

Keywords

• nonn
• tabf
• hard

Comments

Extended to n=1.

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

T(n,n^2) = 1.

Examples

			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
		

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.

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).