%I #17 Oct 27 2024 05:23:45
%S 1,1,1,3,78,265764,71095150000,1791180894322714050,
%T 5782448787773327324756945480,
%U 3129263982719398327413605440811466528960,359946241929477448403523246795300556527932947115524992
%N Number of different (unlabeled) 2-cell embeddings of the complete graph K_n on n vertices into orientable surfaces.
%H B. P. Mull, R. G. Rieper and A. T. White, <a href="https://doi.org/10.1090/S0002-9939-1988-0938690-1">Enumerating 2-cell imbeddings of connected graphs</a>, Proc. Amer. Math. Soc. 103 (1988), 321-330.
%H A. T. White, <a href="https://doi.org/10.1017/S0963548300001395">An introduction to random topological graph theory</a>, Comb., Probab. Comput. 3 (1994), 545-555.
%F a(n) = Sum_{d|n} (n-2)!^(n/d) / (d^(n/d)*(n/d)!) + Sum_{d|n-1, d>1} phi(d) * (n-2)!^((n-1)/d)) / (n-1), n>1, where phi(n) is the Euler totient function A000010.
%F a(n) ~ n^(n^2 - 5*n/2 - 1/2) * (2*Pi)^((n-1)/2) / exp(n*(n-1) - 13/12). - _Vaclav Kotesovec_, Oct 27 2024
%t f[n_] := Block[{d = Divisors[n], d1 = Drop[ Divisors[n - 1], 1]}, Apply[ Plus, (n - 2)!^(n/d)/(d^(n/d)*(n/d)!)] + Apply[ Plus, EulerPhi[d1]*(n - 2)!^((n - 1)/d1)/(n - 1)]]; f[1] = 1; Table[ f[n], {n, 2, 13}]
%Y Cf. A069840.
%K nonn
%O 1,4
%A _Valery A. Liskovets_, Apr 22 2002
%E Edited and extended by _Robert G. Wilson v_ and _Vladeta Jovovic_, May 04 2002