A113949 -id:A113949

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A113947

Number of non-equivalent n-fold branched coverings of the projective plane with two cyclic branch points.

+10
3

1, 2, 3, 8, 16, 64, 264, 1580, 10648, 84320, 750380, 7455312, 81566928, 974988768, 12636692720, 176505029160, 2642791002368, 42224138928712, 716984262871596, 12893605560786944

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..450

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.

FORMULA

a(n) = (1/n)*Sum_{k|n} gcd(2, n/k)*phi(n/k)^2*(n/k)^(k-1) * Sum_{i=0..k-1} i!*(k-i-1)! where phi(n) is the Euler function A000010.

a(n) ~ 2*n!/n^2. - Vaclav Kotesovec, Oct 27 2024

MATHEMATICA

a[n_] := 1/n DivisorSum[n, GCD[2, n/#]*EulerPhi[n/#]^2*(n/#)^(#-1) Sum[i! * (#-i- 1)!, {i, 0, #-1}]&]; Array[a, 20] (* Jean-François Alcover, Oct 05 2016 *)

CROSSREFS

Cf. A113948, A113949.

KEYWORD

nonn

AUTHOR

Valery A. Liskovets, Nov 10 2005

STATUS

approved

A113950

Number of n-fold branched coverings of the Klein bottle with r cyclic branch points (n,r>=1); array read by downward antidiagonals.

+10
2

1, 1, 0, 1, 4, 5, 1, 0, 13, 0, 1, 4, 23, 104, 44, 1, 0, 49, 0, 1256, 0, 1, 4, 95, 2720, 27344, 30608, 1266, 1, 0, 193, 0, 666656, 0, 1071540, 0, 1, 4, 383, 93824, 15911744, 415444544, 743214744

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

The odd bisection of the first column is A113948.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..43.

FORMULA

E.g. for n=7 and r>=1, a(7, r)=2*720^r+(-1)^r*2*120^r+2*48^r+(-1)^r*36^r+6^r (more generally, a(7, r, h)=7^(h-2)*(2*720^m+(-1)^r*2*120^m+2*48^m+(-1)^r*36^m+6^r) for 7-sheeted coverings of the non-orientable surface of genus h>=1, where m=h+r-2).

EXAMPLE

The array begins:

1 1 1 1 1 1 ...

0 4 0 4 0 4 ...

5 13 23 49 95 193 ...

0 104 0 2720 0 93824 ...