A234616 Numbers of undirected cycles in the complete tripartite graph K_{n,n,n}.

1, 63, 6705, 1960804, 1271288295, 1541975757831, 3135880743480163, 9904953891455450640, 45915662047529291081589, 299038026557168514632822455, 2642895689915240835222121682301, 30814273315381549790551229559722628

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..100 (terms 1..50 from Andrew Howroyd)

Andrew Howroyd, Formula for the number of cycles

Eric Weisstein's World of Mathematics, Complete Tripartite Graph

Eric Weisstein's World of Mathematics, Graph Cycle

Formula

Row sums of A296546.

a(n) ~ sqrt(3*Pi) * 2^(3*n - 1/2) * n^(3*n - 1/2) / exp(3*n - 3/2). - Vaclav Kotesovec, Feb 17 2024

Mathematica

Table[(Sum[Binomial[n, k] Binomial[n, i + p] Binomial[n, j + p] Binomial[k, i] Binomial[k - i, j] (k - 1)! (i + p)! (j + p)! 2^(k - i - j) Binomial[p + i + j - 1, k - 1], {k, n}, {i, 0, k}, {j, 0, k - i}, {p, k - i - j, n}] + Sum[Binomial[n, k]^2 k! (k - 1)!, {k, 2, n}])/2 - n^2, {n, 10}] (* Eric W. Weisstein, May 26 2017 *)
Table[(n^2 (HypergeometricPFQ[{1, 1, 1 - n, 1 - n}, {2}, 1] - 3) + Sum[2^(k - i - j) Binomial[k, i] Binomial[k - i, j] Binomial[n, k] Binomial[n, i + p] Binomial[n, j + p] Binomial[i + j + p - 1, k - 1] (k - 1)! (i + p)! (j + p)!, {k, n}, {i, 0, k}, {j, 0, k - i}, {p, k - i - j, n}])/2, {n, 10}] (* Eric W. Weisstein, May 25 2023 *)

Prog

(PARI)
c(n, k, i, j, p) = {binomial(n, k)*binomial(n, i+p)*binomial(n, j+p)*binomial(k, i)*binomial(k-i, j)*(k-1)!*(i+p)!*(j+p)!*2^(k-i-j)*binomial(p+i+j-1, k-1)}
a(n)={(sum(k=1, n, sum(i=0, k, sum(j=0, k-i, sum(p=k-i-j, n, c(n, k, i, j, p) )))) + sum(k=2, n, binomial(n, k)^2*k!*(k-1)!))/2 - n^2} \\ Andrew Howroyd, May 25 2017
(Python)
from sympy import binomial, factorial
def c(n, k, i, j, p): return binomial(n, k)*binomial(n, i + p)*binomial(n, j + p)*binomial(k, i)*binomial(k - i, j)*factorial(k - 1)*factorial(i + p)*factorial(j + p)*2**(k - i - j)*binomial(p + i + j - 1, k - 1)
def a(n): return (sum([sum([sum([sum([c(n, k, i, j, p) for p in range(k - i - j, n + 1)]) for j in range(k - i + 1)]) for i in range(k + 1)]) for k in range(1, n + 1)]) + sum(binomial(n, k)**2*factorial(k)*factorial(k - 1) for k in range(2, n + 1)))/2 - n**2
print([a(k) for k in range(1, 13)]) # Indranil Ghosh, Aug 14 2017, after PARI

Crossrefs

Cf. A234365, A234633, A070968.

Cf. A296546 (cycle polynomial coefficients of K_n,n,n).

Sequence in context: A296782 A292782 A251011 * A093263 A069433 A178634

Adjacent sequences: A234613 A234614 A234615 * A234617 A234618 A234619

Keyword

nonn

Author

Eric W. Weisstein, Dec 28 2013

Extensions

a(7)-a(12) from Andrew Howroyd, May 25 2017

Status

approved