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%I #15 Jan 04 2024 18:10:17
%S 0,4,54,918,31232,2059624,266734812,68574627036,35160753222400,
%T 36021330363615408,73782362964470935112,302225854825997535378632,
%U 2475866675779140063716682240,40564755806137338166417907530592,1329227401912999475682655581004557840
%N Number of total dominating sets in the n-triangular (Johnson) graph.
%H Andrew Howroyd, <a href="/A303048/b303048.txt">Table of n, a(n) for n = 2..50</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*(2*k-1)!!*A290847(n-2*k). - _Andrew Howroyd_, Apr 20 2018
%t b[n_] := Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}]
%t Table[Sum[(-1)^k Binomial[n, 2 k] (2 k)!/(2^k k!) (b[n - 2 k] + (n - 2 k) b[n - 2 k - 1]), {k, 0, Floor[n/2]}], {n, 2, 20}]
%o (PARI) \\ here b(n) is A006129
%o b(n)=sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
%o a(n)=sum(k=0, n\2, (-1)^k*binomial(n,2*k)*(2*k)!/(2^k*k!)*(b(n-2*k) + (n-2*k)*b(n-2*k-1))); \\ _Andrew Howroyd_, Apr 20 2018
%Y Cf. A006129, A290847, A298104, A303227, A304561.
%K nonn
%O 2,2
%A _Eric W. Weisstein_, Apr 17 2018
%E a(8)-a(16) from _Andrew Howroyd_, Apr 20 2018
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