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A290847 Number of dominating sets in the n-triangular graph.

%I A290847 #27 Feb 16 2025 08:33:50
%S A290847 1,7,57,973,32057,2079427,267620753,68649126489,35172776136145,
%T A290847 36025104013571583,73784683970720501897,302228664636911612364581,
%U A290847 2475873390079769597467385417,40564787539999607393632514635067,1329227699017403425105119604848703905
%N A290847 Number of dominating sets in the n-triangular graph.
%C A290847 A dominating set on the triangular graph corresponds with an edge cover on the complete graph with optionally one vertex removed.
%H A290847 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>
%H A290847 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H A290847 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%F A290847 a(n) = A006129(n) + n * A006129(n-1).
%F A290847 a(n) = 2^binomial(n,2) - Sum_{k=2..n} binomial(n,k)*A006129(n-k).
%t A290847 b[n_]:=Sum[(-1)^(n - k)*Binomial[n, k]*2^Binomial[k, 2], {k, 0, n}]; a[n_]:=b[n] + n*b[n - 1]; Table[a[n], {n, 2, 20}] (* _Indranil Ghosh_, Aug 12 2017 *)
%o A290847 (PARI) \\ here b(n) is A006129
%o A290847 b(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
%o A290847 a(n) = b(n) + n*b(n-1);
%o A290847 (Python)
%o A290847 from sympy import binomial
%o A290847 def b(n): return sum((-1)**(n - k)*binomial(n, k)*2**binomial(k, 2) for k in range(n + 1))
%o A290847 def a(n): return b(n) + n*b(n - 1)
%o A290847 print([a(n) for n in range(2, 21)]) # _Indranil Ghosh_, Aug 13 2017
%Y A290847 Cf. A000085, A006129, A193154, A287231, A287689, A290056, A290716.
%K A290847 nonn
%O A290847 2,2
%A A290847 _Andrew Howroyd_, Aug 12 2017