A181950 Weighted sum of all cyclic subgroups of the Alternating Group A_n.

1, 1, 4, 19, 91, 571, 4096, 38599, 370399, 3771751, 40020916, 486887611, 6457566259, 97397627419, 1566407932636, 25622476773391, 416792928270751, 7346982309720079, 141863542111338124, 2968348473040595971, 65223378275792128771, 1460499016109864574691, 32600807940657384926884

Offset

1,3

Comments

Sum of the order of all cyclic subgroups of Alt_n.

Each permutation is counted as many times as it appears in a cyclic subgroup.

a(7) = 2^12 is remarkable as a power of 2.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Formula

a(n) = Sum_{k=1..A051593(n)} k*A303728(n, k). - Andrew Howroyd, Jul 03 2018

Example

a(5) = 1*1 + 2*15 + 3*10 + 5*6 = 1 + 30 +30 +30 = 91.

Prog

(PARI) \\ permcount is number of permutations of given type.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
a(n)={my(s=0); forpart(p=n, if(sum(i=1, #p, p[i]-1)%2==0, my(d=lcm(Vec(p))); s+=d*permcount(p)/eulerphi(d))); s} \\ Andrew Howroyd, Jul 03 2018

Crossrefs

Cf. A051593, A051636, A303728.

Sequence in context: A218988 A151253 A121179 * A288687 A275751 A131552

Adjacent sequences: A181947 A181948 A181949 * A181951 A181952 A181953

Keyword

nonn

Author

Olivier Gérard, Apr 03 2012

Extensions

Terms a(9) and beyond from Andrew Howroyd, Jul 03 2018

Status

approved