%I #15 Jan 08 2022 12:11:10

%S 1,1,5,10,23,40,73,114,180,262,379,521,712,938,1228,1567,1986,2469,

%T 3052,3715,4499,5383,6410,7558,8875,10335,11991,13816,15865,18110,

%U 20611,23336,26350,29620,33213,37095,41338,45904,50870,56197,61964,68131,74782,81873

%N Number of factorizations of m^n into 3 factors, where m is a product of exactly 3 distinct primes and each factor is a product of n primes (counted with multiplicity).

%H Alois P. Heinz, <a href="/A257464/b257464.txt">Table of n, a(n) for n = 0..1000</a>

%H Alexander V. Karpov, <a href="https://wp.hse.ru/data/2018/04/04/1164595187/188EC2018.pdf">An Informational Basis for Voting Rules</a>, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.

%H P. A. MacMahon, <a href="http://plms.oxfordjournals.org/content/s2-17/1/25.extract">Combinations derived from m identical sets of n different letters and their connexion with general magic squares</a>, Proc. London Math. Soc., 17 (1917), 25-41. This sequence is column 3 of table f(m,n) on page 40.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,-1,1,3,-1,-2,1).

%F G.f.: -(x^6-x^5+2*x^4+2*x^3+2*x^2-x+1)/((x^2+x+1)*(x+1)^2*(x-1)^5).

%e a(2) = 5: (2*3*5)^2 = 900 = 10*10*9 = 15*10*6 = 15*15*4 = 25*6*6 = 25*9*4.

%e a(4) = 23: (2*3*5)^4 = 810000 = 100*90*90 = 100*100*81 = 135*100*60 = 150*90*60 = 150*100*54 = 150*135*40 = 150*150*36 = 225*60*60 = 225*90*40 = 225*100*36 = 225*150*24 = 225*225*16 = 250*60*54 = 250*81*40 = 250*90*36 = 250*135*24 = 375*54*40 = 375*60*36 = 375*90*24 = 375*135*16 = 625*36*36 = 625*54*24 = 625*81*16.

%p a:= n-> coeff(series(-(x^6-x^5+2*x^4+2*x^3+2*x^2-x+1)/

%p ((x^2+x+1)*(x+1)^2*(x-1)^5), x, n+1), x, n):

%p seq(a(n), n=0..60);

%t CoefficientList[Series[-(x^6 - x^5 + 2 x^4 + 2 x^3 + 2 x^2 - x + 1)/((x^2 + x + 1) (x + 1)^2*(x - 1)^5), {x, 0, 43}], x] (* _Michael De Vlieger_, Jul 02 2018 *)

%t LinearRecurrence[{2,1,-3,-1,1,3,-1,-2,1},{1,1,5,10,23,40,73,114,180},50] (* _Harvey P. Dale_, Jan 08 2022 *)

%Y Row n=3 of A257463.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Apr 24 2015