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A290847
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Number of dominating sets in the n-triangular graph.
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4
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1, 7, 57, 973, 32057, 2079427, 267620753, 68649126489, 35172776136145, 36025104013571583, 73784683970720501897, 302228664636911612364581, 2475873390079769597467385417, 40564787539999607393632514635067, 1329227699017403425105119604848703905
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OFFSET
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2,2
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COMMENTS
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A dominating set on the triangular graph corresponds with an edge cover on the complete graph with optionally one vertex removed.
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LINKS
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FORMULA
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a(n) = 2^binomial(n,2) - Sum_{k=2..n} binomial(n,k)*A006129(n-k).
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MATHEMATICA
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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 *)
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PROG
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b(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
a(n) = b(n) + n*b(n-1);
(Python)
from sympy import binomial
def b(n): return sum((-1)**(n - k)*binomial(n, k)*2**binomial(k, 2) for k in range(n + 1))
def a(n): return b(n) + n*b(n - 1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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