A378413 Irregular triangle read by rows: T(n,k) is the coefficient of x^k in the domination polynomial of the n-prism graph (n>=1, A004524(n+2)<=k<=2*n).
2, 1, 6, 4, 1, 9, 20, 15, 6, 1, 4, 24, 62, 56, 28, 8, 1, 10, 85, 192, 200, 120, 45, 10, 1, 51, 288, 618, 696, 483, 220, 66, 12, 1, 14, 210, 966, 2018, 2408, 1862, 987, 364, 91, 14, 1, 4, 80, 824, 3248, 6646, 8304, 6992, 4176, 1804, 560, 120, 16, 1, 18, 405
Offset: 1
Examples
D(1) = 2*x+x^2 D(2) = 6*x^2+4*x^3+x^4 D(3) = 9*x^2+20*x^3+15*x^4+6*x^5+x^6 D(4) = 4*x^2+24*x^3+62*x^4+56*x^5+28*x^6+8*x^7+x^8 D(5) = 10*x^3+85*x^4+192*x^5+200*x^6+120*x^7+45*x^8+10*x^9+x^10
Links
- Eric W. Weisstein, Table of n, a(n) for n = 1..462
- 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, Prism Graph.
Crossrefs
Programs
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Mathematica
DeleteCases[#, 0] & /@ CoefficientList[LinearRecurrence[{2 x + x^2, x^2, 3 x^2 + 2 x^3, x^2, x^4, -x^4, -x^4}, {2 x + x^2, 6 x^2 + 4 x^3 + x^4, 9 x^2 + 20 x^3 + 15 x^4 + 6 x^5 + x^6, 4 x^2 + 24 x^3 + 62 x^4 + 56 x^5 + 28 x^6 + 8 x^7 + x^8, 10 x^3 + 85 x^4 + 192 x^5 + 200 x^6 + 120 x^7 + 45 x^8 + 10 x^9 + x^10, 51 x^4 + 288 x^5 + 618 x^6 + 696 x^7 + 483 x^8 + 220 x^9 + 66 x^10 + 12 x^11 + x^12, 14 x^4 + 210 x^5 + 966 x^6 + 2018 x^7 + 2408 x^8 + 1862 x^9 + 987 x^10 + 364 x^11 + 91 x^12 + 14 x^13 + x^14}, 10], x] // Flatten
Formula
D(n) = (2*x+x^2)*D(n-1) + x^2*D(n-2) + (3*x^2+2x^3)*D(n-3) + x^2*D(n-4) + x^4*D(n-5) - x^4*D(n-6) -x^4*D(n-7), where D(n) = sum(T(n,k)*x^k,k).
Comments