A113947 -id:A113947

Search: a113947 -id:a113947

Displaying 1-2 of 2 results found.

page 1

A113948

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

+10
2

1, 5, 44, 1266, 72636, 6652810, 889574412, 163459302788, 39520825344016, 12164510040883218, 4644631106520877974, 2154334728240414720022, 1193170003333152768100020, 777776389315596583864343748

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

OFFSET

0,2

COMMENTS

No such covering of even multiplicity exists.

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=0..13.

FORMULA

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.

a(n) ~ sqrt(Pi) * 2^(2*n + 2) * n^(2*n + 1/2) / exp(2*n). - Vaclav Kotesovec, Oct 27 2024

MATHEMATICA

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 *)

CROSSREFS

Cf. A113947, A113950.

KEYWORD

nonn

AUTHOR

Valery A. Liskovets, Nov 10 2005

STATUS

approved

A113949

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

+10
2

1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 2, 5, 8, 1, 1, 0, 11, 0, 16, 0, 1, 2, 21, 128, 232, 64, 1, 1, 0, 43, 0, 5680, 0, 264, 0, 1, 2, 85, 3968, 132448, 581696, 144504, 1580, 1, 1, 0, 171, 0, 3189184, 0, 107174448, 0, 10648, 0, 1, 2, 341, 140288, 76426624, 8297164544

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

OFFSET

1,5

COMMENTS

The second column is A113947.

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..61.

FORMULA

E.g. for n=7 and r>=1, a(7, r)=(2*720^(r-1)+(-1)^r*2*120^(r-1)+2*48^(r-1)+(-1)^r*36^(r-1)+6^r)/7 (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 1 ...

0 2 0 2 0 2 0 ...

1 3 5 11 21 43 85 ...

0 8 0 128 0 3968 0 ...