OFFSET
1,2
COMMENTS
a(n) is the number of linear partitions of the linearly ordered set [n] = {1,2,...,n} with blocks of the same size, where each block has two element marked (possibly equal). For instance, for n = 3, we have the following 10 linear partitions (where the marked elements are denoted by a and b, or by X when they coincide):
(X)(X)(X), (ab3), (a2b), (1ab), (ba3), (b2a), (1ba), (X23), (1X3), (12X). - Emanuele Munarini, Feb 03 2014
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..3143
FORMULA
G.f.: Sum_{n>=1} -log(1 - (n^2)*x^n)/n = Sum_{n>=1} a(n) x^n/n.
G.f.: Sum_{k>=1} k^2*x^k/(1-k^2*x^k). - Benoit Cloitre, Apr 21 2003
EXAMPLE
a(10) = (10/1)^(2*1) +(10/2)^(2*2) +(10/5)^(2*5) +(10/10)^(2*10) = 1750 because positive divisors of 10 are 1, 2, 5, 10.
MATHEMATICA
Table[Total[Quotient[n, x = Divisors[n]]^(2*x)], {n, 34}] (* Jayanta Basu, Jul 08 2013 *)
PROG
(PARI) a(n)=sumdiv(n, d, (d)^(2*n/d) ); /* Joerg Arndt, Oct 07 2012 */
(Maxima) a(n):= lsum(d^(2*n/d), d, listify(divisors(n)));
makelist(a(n), n, 1, 40); /* Emanuele Munarini , Feb 03 2014 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul D. Hanna, Aug 04 2002
EXTENSIONS
Corrected a(14) and inserted missing a(16) by Jayanta Basu, Jul 08 2013
STATUS
approved