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A378241 Numbers of directed Hamiltonian cycles in the complete 4-partite graph K_{n,n,n,n}.

%I A378241 #25 Feb 20 2025 10:31:25
%S A378241 6,1488,3667680,37744330752,1106491456512000,74213488705904640000,
%T A378241 9872975878366503813120000,2355966665497190945783808000000,
%U A378241 935825492908108988335792827924480000,584053924678169568704863421815848960000000
%N A378241 Numbers of directed Hamiltonian cycles in the complete 4-partite graph K_{n,n,n,n}.
%H A378241 P. Horak and L. Tovarek, <a href="https://dml.cz/handle/10338.dmlcz/132268">On Hamiltonian cycles of complete n-partite graphs</a>, Mathematica Slovaca, vol. 29 (1979), 43-47.
%H A378241 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Completek-PartiteGraph.html">Complete k-Partite Graph</a>.
%H A378241 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>.
%F A378241 a(n) = 3!*(n-1)!*(n!)^3*A369923(n,4). - _Andrew Howroyd_, Nov 20 2024
%F A378241 a(n) = 2*A381326(n). - _Eric W. Weisstein_, Feb 20 2025
%t A378241 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}]
%t A378241 Table[(n!)^4 Expand[Hypergeometric1F1[1 - n, 2, x]^4 x^3] /. x^p_ :> p!, {n, 10}] (* _Eric W. Weisstein_, Feb 20 2025 *)
%o A378241 (Python)
%o A378241 from math import factorial as fact
%o A378241 def a(n):
%o A378241    # Using formula found in Horak et al.
%o A378241    return sum(sum(sum(sum(
%o A378241        fact(2*n-i-j-1)*pow(2,2*i)*pow(3,j)*pow(fact(n),4)//fact(j) *
%o A378241        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)) *
%o A378241        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))
%o A378241        for e in range(max(0,i+j-n+d), min(d,2*n-2*i-2*j-d)+1))
%o A378241        for d in range(max(0,n-j-2*i), min(n-j,2*n-3*i-2*j)+1))
%o A378241        for i in range(int(2*(n-j)/3)+1))
%o A378241        for j in range(n+1))
%o A378241 print([a(n) for n in range(1,11)])
%Y A378241 Cf. A209183, A234365, A369923, A377586.
%K A378241 nonn
%O A378241 1,1
%A A378241 _Zlatko Damijanic_, Nov 20 2024