%I #6 Oct 27 2024 05:00:10

%S 1,5,44,1266,72636,6652810,889574412,163459302788,39520825344016,

%T 12164510040883218,4644631106520877974,2154334728240414720022,

%U 1193170003333152768100020,777776389315596583864343748

%N Number of non-equivalent (2n+1)-fold branched coverings of the Klein bottle with one cyclic branch point.

%C No such covering of even multiplicity exists.

%D J. H. Kwak, A. Mednykh and V. Liskovets, Enumeration of branched coverings of nonorientable surfaces with cyclic branch points, SIAM J. Discrete Math., Vol. 19, No. 2 (2005), 388-398.

%F a(n)=2*sum_{k|(2n+1)}k!*((2n+1)/k)^(k-1)*phi((2n+1)/k)/(k+1) where phi(n) is the Euler function A000010.

%F a(n) ~ sqrt(Pi) * 2^(2*n + 2) * n^(2*n + 1/2) / exp(2*n). - _Vaclav Kotesovec_, Oct 27 2024

%t Table[2*Sum[k!*((2*n + 1)/k)^(k-1) * EulerPhi[(2*n + 1)/k] / (k+1), {k, Divisors[2*n + 1]}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 27 2024 *)

%Y Cf. A113947, A113950.

%K nonn

%O 0,2

%A _Valery A. Liskovets_, Nov 10 2005