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A234616 Numbers of undirected cycles in the complete tripartite graph K_{n,n,n}. 5
1, 63, 6705, 1960804, 1271288295, 1541975757831, 3135880743480163, 9904953891455450640, 45915662047529291081589, 299038026557168514632822455, 2642895689915240835222121682301, 30814273315381549790551229559722628 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..50

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.

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 *)

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

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Last modified August 7 16:11 EDT 2020. Contains 336276 sequences. (Running on oeis4.)