This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378413 #17 Nov 26 2024 09:32:15
%S A378413 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,
%T A378413 288,618,696,483,220,66,12,1,14,210,966,2018,2408,1862,987,364,91,14,
%U A378413 1,4,80,824,3248,6646,8304,6992,4176,1804,560,120,16,1,18,405
%N 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).
%C A378413 Sequence extended to n=1 using the recurrence.
%C A378413 Sum_{k=A004524(n+2)..2*n} T(n,k) = A284702(n).
%C A378413 T(n,2*n) = 1.
%H A378413 Eric W. Weisstein, <a href="/A378413/b378413.txt">Table of n, a(n) for n = 1..462</a>
%H A378413 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 A378413 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>.
%H A378413 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>.
%H A378413 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominationPolynomial.html">Domination Polynomial</a>.
%H A378413 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>.
%F A378413 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).
%e A378413 D(1) = 2*x+x^2
%e A378413 D(2) = 6*x^2+4*x^3+x^4
%e A378413 D(3) = 9*x^2+20*x^3+15*x^4+6*x^5+x^6
%e A378413 D(4) = 4*x^2+24*x^3+62*x^4+56*x^5+28*x^6+8*x^7+x^8
%e A378413 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
%t A378413 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
%Y A378413 Cf. A004524 (domination number of the (n-2)-prism graph).
%Y A378413 Cf. A284702 (number of dominating sets in the n-prism graph).
%Y A378413 Cf. A005843 (vertex count of the n-prism graph = 2*n).
%K A378413 nonn,tabf
%O A378413 1,1
%A A378413 _Eric W. Weisstein_, Nov 25 2024