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A129348 Number of (directed) Hamiltonian circuits in the cocktail party graph of order n. 4
0, 2, 32, 1488, 112512, 12771840, 2036229120, 434469611520, 119619533537280, 41303040523960320, 17481826772405452800, 8902337068174698086400, 5370014079716477003366400, 3786918976243761421064601600, 3087031512410698159166482022400, 2880726660365605475506018320384000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also, the number of ways (up to rotations) to sit n married couples at a circular table with no spouses next to each other.  Cf. A007060, A193639. - Geoffrey Critzer, Feb 09 2014

The cocktail party graph may also be called the n-octohedron, n-orthoplex or n dimensional cross polytope. - Andrew Howroyd, May 14 2017

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..100

Marko R. Riedel, Math.Stackexchange.com Proof of asymptotic (saddle point method) and closed form (inclusion-exclusion)

Eric Weisstein's World of Mathematics, Cocktail Party Graph

Eric Weisstein's World of Mathematics, Hamiltonian Cycle

FORMULA

For n>=2, a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*(2*n-1-k)!*2^k. - Geoffrey Critzer, Feb 09 2014

Recurrence (for n>=4): (2*n-3)*a(n) = 2*(n-1)*(4*n^2 - 8*n + 5)*a(n-1) + 4*(n-2)*(n-1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Feb 09 2014

a(n) ~ sqrt(Pi) * 2^(2*n) * n^(2*n-1/2) / exp(2*n+1). - Vaclav Kotesovec, Feb 09 2014

For n>=2, a(n) = (-1 + 2 n)! Hypergeometric1F1[-n, 1 - 2 n, -2]. - Eric W. Weisstein, Mar 29 2014

a(n) = A003435(n) / (2*n) = A003436(n) * (n-1)! * 2^(n-1). - Andrew Howroyd, May 14 2017

MAPLE

a:= proc(n) option remember; `if`(n<3, n*(n-1),

     ((136*n^3-608*n^2+762*n-470) *a(n-1)

       +4*(n-2)*(14*n^2+29*n-193) *a(n-2)

       -80*(n-2)*(n-3)*(n-4) *a(n-3)) /(34*n-101))

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, Feb 09 2014

MATHEMATICA

Prepend[Table[Sum[(-1)^i Binomial[n, i] (2n - 1 - i)! 2^i, {i, 0, n}], {n, 2, 16}], 0] (* Geoffrey Critzer, Feb 09 2014 *)

Table[Piecewise[{{(-1 + 2 n)! Hypergeometric1F1[-n, 1 - 2 n, -2],

    n > 1}}], {n, 16}] (* Eric W. Weisstein, Mar 29 2014 *)

PROG

(PARI) { A129348(n) = sum(m=0, n-1, sum(k=1, n-m, (-1)^k * binomial(n-1, m) * binomial(n-m-1, k-1) * 2^(k-1) * ([0, k-1, 2*(n-m-k); 1, k-2, 2*(n-m-k); 1, k-1, 2*(n-m-k-1)]^(2*n))[1, 1] ) + sum(k=0, n-m, (-1)^k * binomial(n-1, m) * binomial(n-m-1, k) * 2^(k-1) * ([0, k, 2*(n-m-k-1); 2, k-1, 2*(n-m-k-1); 2, k, 2*(n-m-k-2)]^(2*n))[1, 1] ) ) } \\ Max Alekseyev, Dec 22 2013

CROSSREFS

Cf. A003435, A003436, A003437, A007060, A167987.

Sequence in context: A012209 A295418 A172286 * A280211 A087084 A193269

Adjacent sequences:  A129345 A129346 A129347 * A129349 A129350 A129351

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Apr 10 2007

EXTENSIONS

Terms a(6) onward from Max Alekseyev, Nov 10 2007

STATUS

approved

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Last modified December 18 12:06 EST 2018. Contains 318229 sequences. (Running on oeis4.)