A378412 Irregular triangle read by rows: T(n,k) is the coefficient of x^k in the domination polynomial of the n X n grid graph (n>=1, A104519(n+2)<=k<=n^2).

1, 6, 4, 1, 10, 57, 98, 80, 36, 9, 1, 2, 40, 554, 2484, 5494, 7268, 6402, 3964, 1760, 556, 120, 16, 1, 22, 1545, 22594, 140304, 492506, 1126091, 1823057, 2204694, 2063202, 1528544, 908623, 435832, 168426, 51953, 12550, 2296, 300, 25, 1, 288, 20896, 478624

Offset

1,2

Comments

Sum_{k=A104519(n+2)..n^2} T(n,k) = A133515(n).

T(n,n^2) = 1.

Links

Eric W. Weisstein, Table of n, a(n) for n = 1..2922

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, Domatic Number.

Eric Weisstein's World of Mathematics, Dominating Set.

Eric Weisstein's World of Mathematics, Domination Number.

Eric Weisstein's World of Mathematics, Grid Graph.

Example

D_1(x)=x
D_2(x)=6*x^2+4*x^3+x^4
D_3(x)=10*x^3+57*x^4+98*x^5+80*x^6+36*x^7+9*x^8+x^9
D_4(x)=2*x^4+40*x^5+554*x^6+2484*x^7+5494*x^8+7268*x^9+6402*x^10+3964*x^11+1760*x^12+556*x^13+120*x^14+16*x^15+x^16

Crossrefs

Cf. A104519 (domination number of the (n-2) X (n-2) grid graph).

Cf. A133515 (number of dominating sets in the n X n grid graph).

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

Sequence in context: A365319 A343614 A086241 * A204023 A360984 A378418

Adjacent sequences: A378409 A378410 A378411 * A378413 A378414 A378415

Keyword

nonn,tabf,hard

Author

Eric W. Weisstein, Nov 25 2024

Status

approved