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

Author

Alois P. Heinz, Apr 27 2015

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Row n=2 of A257462.

Cf. A002426, A097861.

Programs

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);

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.