A257520 Number of factorizations of m^2 into 2 factors, where m is a product of exactly n distinct primes and each factor is a product of n primes (counted with multiplicity).

1, 1, 2, 4, 10, 26, 71, 197, 554, 1570, 4477, 12827, 36895, 106471, 308114, 893804, 2598314, 7567466, 22076405, 64498427, 188689685, 552675365, 1620567764, 4756614062, 13974168191, 41088418151, 120906613076, 356035078102, 1049120176954, 3093337815410

Offset

0,3

Comments

Also number of ways to partition the multiset consisting of 2 copies each of 1, 2, ..., n into 2 multisets of size n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1/sqrt((1+x)*(1-3*x))+1/(1-x))/2.

E.g.f.: exp(x)*(1+BesselI(0,2*x))/2.

a(n) = ((3*n^2-7*n+3)*a(n-1) +(n-1)*(n-3)*a(n-2) -3*(n-1)*(n-2)*a(n-3)) / (n*(n-2)) for n>2, a(0) = a(1) = 1, a(2) = 2.

a(n) = (A002426(n)+1)/2.

a(n) = A097861(n)+1.

Example

a(4) = 10: (2*3*5*7)^2 = 44100 = 210*210 = 225*196 = 294*150 = 315*140 = 350*126 = 441*100 = 490*90 = 525*84 = 735*60 = 1225*36.

Maple

a:= proc(n) option remember; `if`(n<3, [1, 1, 2][n+1],
      ((3*n^2-7*n+3)*a(n-1) +(n-1)*(n-3)*a(n-2)
       -3*(n-1)*(n-2)*a(n-3)) / (n*(n-2)))
    end:
seq(a(n), n=0..40);

Crossrefs

Row n=2 of A257462.

Cf. A002426, A097861.

Sequence in context: A055226 A097085 A071962 * A035079 A259206 A106221

Adjacent sequences: A257517 A257518 A257519 * A257521 A257522 A257523

Keyword

nonn

Author

Alois P. Heinz, Apr 27 2015

Status

approved