A027844 Number of subgroups of index n of fundamental group of the non-orientable cycle bundle over the Klein bottle.

1, 7, 7, 27, 11, 55, 15, 91, 34, 97, 23, 231, 27, 147, 77, 299, 35, 334, 39, 437, 105, 271, 47, 847, 86, 345, 142, 699, 59, 865, 63, 1003, 161, 517, 165, 1590, 75, 615, 189, 1701, 83, 1371, 87, 1391, 374, 835, 95, 3023, 162, 1322, 245, 1821, 107, 2062, 253, 2835

Offset

1,2

Comments

From a recent general formula of Stanley's for the number of subgroups in G\times Z.

References

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64.

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..20000

G. Chelnokov, M. Deryagina, A. Mednykh, On the Coverings of Amphicosms; Revised title: On the coverings of Euclidian manifolds B_1 and B_2, arXiv preprint arXiv:1502.01528 [math.AT], 2015.

V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Formula

Sum k*b(k), k|n, where b(k) is the number of n-list coverings of the Klein bottle (A046524).

Mathematica

b[k_] := If[OddQ[k], DivisorSigma[0, k], (3 DivisorSigma[0, k] + DivisorSigma[1, k/2] - DivisorSigma[0, k/2])/2]; a[n_] := Sum[k*b[k], {k, Divisors[n]}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jul 19 2012 *)

Prog

(PARI)
A001001(n) = sumdiv(n, d, sigma(d) * d);
A060640(n) = sumdiv(n, d, sigma(n\d) * d);
S1(n)      = if (n%2, 0, A001001(n\2));
S11(n)     = A060640(n) - if(n%2, 0, A060640(n\2));
S21(n)     = if (n%2, 0, 2*A060640(n\2)) - if (n%4, 0, 2*A060640(n\4));
a(n) = S1(n) + S11(n) + S21(n);
vector(56, n, a(n))  \\ Gheorghe Coserea, May 05 2016

Crossrefs

Cf. A001001, A027845, A046524.

Sequence in context: A295733 A341201 A255277 * A268867 A111217 A339339

Adjacent sequences: A027841 A027842 A027843 * A027845 A027846 A027847

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Valery A. Liskovets

Corrected and extended by Vladeta Jovovic, Feb 03 2003

Status

approved