A378418 - OEIS

%I #7 Nov 26 2024 09:32:06

%S 1,6,4,1,48,117,126,84,36,9,1,40,560,2736,6800,10310,10560,7832,4352,

%T 1820,560,120,16,1,10,200,3050,31525,188700,677690,1610700,2740775,

%U 3527075,3562700,2895610,1923600,1053175,475950,176600,53105,12650,2300,300,25,1,18

%N 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).

%C Extended to n=1.

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

%C T(n,n^2) = 1.

%H Eric W. Weisstein, <a href="/A378418/b378418.txt">Table of n, a(n) for n = 1..175</a>

%H Stephan Mertens, <a href="https://arxiv.org/abs/2408.08053">Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph</a>, arXiv:2408.08053 [math.CO], Aug 2024.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominationPolynomial.html">Domination Polynomial</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>.

%e D(1) = x

%e D(2)= 6*x^2+4*x^3+x^4

%e D(3) = 48*x^3+117*x^4+126*x^5+84*x^6+36*x^7+9*x^8+x^9

%e 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

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

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

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

%K nonn,tabf,hard

%O 1,2

%A _Eric W. Weisstein_, Nov 25 2024