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A257464 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).

%I A257464 #15 Jan 08 2022 12:11:10
%S A257464 1,1,5,10,23,40,73,114,180,262,379,521,712,938,1228,1567,1986,2469,
%T A257464 3052,3715,4499,5383,6410,7558,8875,10335,11991,13816,15865,18110,
%U A257464 20611,23336,26350,29620,33213,37095,41338,45904,50870,56197,61964,68131,74782,81873
%N A257464 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 A257464 Alois P. Heinz, <a href="/A257464/b257464.txt">Table of n, a(n) for n = 0..1000</a>
%H A257464 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 A257464 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 A257464 <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 A257464 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 A257464 a(2) = 5: (2*3*5)^2 = 900 = 10*10*9 = 15*10*6 = 15*15*4 = 25*6*6 = 25*9*4.
%e A257464 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 A257464 a:= n-> coeff(series(-(x^6-x^5+2*x^4+2*x^3+2*x^2-x+1)/
%p A257464         ((x^2+x+1)*(x+1)^2*(x-1)^5), x, n+1), x, n):
%p A257464 seq(a(n), n=0..60);
%t A257464 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 A257464 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 A257464 Row n=3 of A257463.
%K A257464 nonn,easy
%O A257464 0,3
%A A257464 _Alois P. Heinz_, Apr 24 2015