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A303048 Number of total dominating sets in the n-triangular (Johnson) graph.

%I A303048 #16 Feb 16 2025 08:33:53
%S A303048 0,4,54,918,31232,2059624,266734812,68574627036,35160753222400,
%T A303048 36021330363615408,73782362964470935112,302225854825997535378632,
%U A303048 2475866675779140063716682240,40564755806137338166417907530592,1329227401912999475682655581004557840
%N A303048 Number of total dominating sets in the n-triangular (Johnson) graph.
%H A303048 Andrew Howroyd, <a href="/A303048/b303048.txt">Table of n, a(n) for n = 2..50</a>
%H A303048 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H A303048 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>
%H A303048 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%F A303048 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 A303048 b[n_] := Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}]
%t A303048 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 A303048 (PARI) \\ here b(n) is A006129
%o A303048 b(n)=sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2));
%o A303048 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 A303048 Cf. A006129, A290847, A298104, A303227, A304561.
%K A303048 nonn
%O A303048 2,2
%A A303048 _Eric W. Weisstein_, Apr 17 2018
%E A303048 a(8)-a(16) from _Andrew Howroyd_, Apr 20 2018