cp's OEIS Frontend

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).

%I A257520 #18 Apr 29 2015 12:36:12
%S A257520 1,1,2,4,10,26,71,197,554,1570,4477,12827,36895,106471,308114,893804,
%T A257520 2598314,7567466,22076405,64498427,188689685,552675365,1620567764,
%U A257520 4756614062,13974168191,41088418151,120906613076,356035078102,1049120176954,3093337815410
%N 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).
%C A257520 Also number of ways to partition the multiset consisting of 2 copies each of 1, 2, ..., n into 2 multisets of size n.
%H A257520 Alois P. Heinz, <a href="/A257520/b257520.txt">Table of n, a(n) for n = 0..1000</a>
%F A257520 G.f.: (1/sqrt((1+x)*(1-3*x))+1/(1-x))/2.
%F A257520 E.g.f.: exp(x)*(1+BesselI(0,2*x))/2.
%F A257520 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.
%F A257520 a(n) = (A002426(n)+1)/2.
%F A257520 a(n) = A097861(n)+1.
%e A257520 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.
%p A257520 a:= proc(n) option remember; `if`(n<3, [1, 1, 2][n+1],
%p A257520       ((3*n^2-7*n+3)*a(n-1) +(n-1)*(n-3)*a(n-2)
%p A257520        -3*(n-1)*(n-2)*a(n-3)) / (n*(n-2)))
%p A257520     end:
%p A257520 seq(a(n), n=0..40);
%Y A257520 Row n=2 of A257462.
%Y A257520 Cf. A002426, A097861.
%K A257520 nonn
%O A257520 0,3
%A A257520 _Alois P. Heinz_, Apr 27 2015