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A369822 Number of (undirected) Eulerian cycles in the (2n)-dipyramid graph.

%I A369822 #23 Sep 06 2025 11:04:54
%S A369822 6,372,68880,26310816,17145457920,17034981004800,23977057921689600,
%T A369822 45400487332999680000,111298452508871250739200,
%U A369822 342962787786595749642240000,1297585985940925048243814400000,5913686127296455213253427855360000,31954282139197508581861513887744000000
%N A369822 Number of (undirected) Eulerian cycles in the (2n)-dipyramid graph.
%C A369822 Sequence extended to a(1) using the formula. - _Eric W. Weisstein_, Sep 06 2025
%H A369822 Andrew Howroyd, <a href="/A369822/b369822.txt">Table of n, a(n) for n = 2..100</a>
%H A369822 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DipyramidalGraph.html">Dipyramidal Graph</a>.
%H A369822 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianCycle.html">Eulerian Cycle</a>.
%F A369822 a(n) = n!*(n-1)!*(2^(2*n)*Sum_{k=0..n} binomial(2*n, 2*k)*binomial(2*k, k) - binomial(2*n, n) - 4*Sum_{q=0..2*n-2} binomial(q, floor(q/2)) * A193858(2*n-2, q)). - _Andrew Howroyd_, Feb 18 2024
%t A369822 Table[n! (n - 1)! (4^n Hypergeometric2F1[1/2 - n, -n, 1, 4] - Binomial[2 n, n] - 4 Sum[2^(2 n - 2) Binomial[2 n - 2, q] Binomial[q, Floor[q/2]] Hypergeometric2F1[1, -q, 2 - 2 n, 1/2], {q, 0, 2 n - 2}]), {n, 20}] (* _Eric W. Weisstein_, Sep 06 2025 *)
%o A369822 (PARI) \\ B(n,k) is A193858(n,k)
%o A369822 B(m,q)={sum(j=0, q, 2^(m-j) * binomial(m-j,q-j))}
%o A369822 a(n)={n!*(n-1)!*(2^(2*n)*sum(k=0, n, binomial(2*n, 2*k)*binomial(2*k, k)) - binomial(2*n, n) - 4*sum(q=0, 2*n-2, binomial(q, q\2) * B(2*n-2, q)))} \\ _Andrew Howroyd_, Feb 18 2024
%Y A369822 Cf. A193858.
%K A369822 nonn,changed
%O A369822 1,1
%A A369822 _Eric W. Weisstein_, Feb 02 2024
%E A369822 a(5) from _Max Alekseyev_, Feb 17 2024
%E A369822 a(6) onwards from _Andrew Howroyd_, Feb 17 2024
%E A369822 a(1) prepended by _Eric W. Weisstein_, Sep 06 2025