OFFSET
1,1
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 = 1..100
Eric Weisstein's World of Mathematics, Dipyramidal Graph.
Eric Weisstein's World of Mathematics, Eulerian Cycle.
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
a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * 3^(2*n - 1/2) * n^(2*n - 1/2) / exp(2*n). - Vaclav Kotesovec, Feb 07 2026
EXAMPLE
a(1) = 6 because the 2-dipyramid graph is the graph {A-B,A-C,B-C,B--C,B-D,C-D} (There are 2 different edges connecting vertex B and C), there are a(1) = 6 Eulerian cycles: {A-B-C--B-D-C-A,A-B--C-B-D-C-A,A-B-D-C--B-C-A,A-B-D-C-B--C-A,A-B-C-D-B--C-A,A-B--C-D-B-C-A}. - Zhuorui He, Oct 30 2025
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Feb 02 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
STATUS
approved