OFFSET
1,1
LINKS
P. Horak and L. Tovarek, On Hamiltonian cycles of complete n-partite graphs, Mathematica Slovaca, vol. 29 (1979), 43-47.
Eric Weisstein's World of Mathematics, Complete k-Partite Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
FORMULA
a(n) = 3!*(n-1)!*(n!)^3*A369923(n,4). - Andrew Howroyd, Nov 20 2024
a(n) = 2*A381326(n). - Eric W. Weisstein, Feb 20 2025
MATHEMATICA
Table[Sum[Sum[Sum[Sum[ (2n - i - j - 1)! 2^(2i) 3^j (n!)^4/(j!) * (3n - 3i - 3j - 2d)!/((2i + j - n + d)! (n - j - d)! (2n - 3i - 2j - d)!) * (2n - 2i - 2j - 2e)!/(e! (d - e)! (2n - 2i - 2j - d - e)! (n - i - j - d + e)! ((n - i - j - e)!)^2), {e, Max[0, i + j - n + d], Min[d, 2n - 2i - 2j - d]}], {d, Max[0, n - j - 2i], Min[n - j, 2n - 3i - 2j]}], {i, 0, Floor[2(n - j)/3]}], {j, 0, n}], {n, 1, 10}]
Table[(n!)^4 Expand[Hypergeometric1F1[1 - n, 2, x]^4 x^3] /. x^p_ :> p!, {n, 10}] (* Eric W. Weisstein, Feb 20 2025 *)
PROG
(Python)
from math import factorial as fact
def a(n):
# Using formula found in Horak et al.
return sum(sum(sum(sum(
fact(2*n-i-j-1)*pow(2, 2*i)*pow(3, j)*pow(fact(n), 4)//fact(j) *
fact(3*n-3*i-3*j-2*d)//(fact(2*i+j-n+d)*fact(n-j-d)*fact(2*n-3*i-2*j-d)) *
fact(2*n-2*i-2*j-2*e)//(fact(e)*fact(d-e)*fact(2*n-2*i-2*j-d-e)*fact(n-i-j-d+e)*pow(fact(n-i-j-e), 2))
for e in range(max(0, i+j-n+d), min(d, 2*n-2*i-2*j-d)+1))
for d in range(max(0, n-j-2*i), min(n-j, 2*n-3*i-2*j)+1))
for i in range(int(2*(n-j)/3)+1))
for j in range(n+1))
print([a(n) for n in range(1, 11)])
CROSSREFS
KEYWORD
nonn
AUTHOR
Zlatko Damijanic, Nov 20 2024
STATUS
approved