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

6, 372, 68880, 26310816, 17145457920, 17034981004800, 23977057921689600, 45400487332999680000, 111298452508871250739200, 342962787786595749642240000, 1297585985940925048243814400000, 5913686127296455213253427855360000, 31954282139197508581861513887744000000

Offset: 1

Author

Eric W. Weisstein, Feb 02 2024

Keywords

• nonn
• changed

Comments

Sequence extended to a(1) using the formula. - Eric W. Weisstein, Sep 06 2025

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..100
• Eric Weisstein's World of Mathematics, Dipyramidal Graph.
• Eric Weisstein's World of Mathematics, Eulerian Cycle.

Crossrefs

Cf. A193858.

Programs

Mathematica

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 *)

PARI

\\ B(n,k) is A193858(n,k)
B(m,q)={sum(j=0, q, 2^(m-j) * binomial(m-j,q-j))}
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

Formula

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

Extensions

a(5) from Max Alekseyev, Feb 17 2024

a(6) onwards from Andrew Howroyd, Feb 17 2024

a(1) prepended by Eric W. Weisstein, Sep 06 2025