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A256493 Number of factorizations of m^3 into at most 3 factors, where m is a product of exactly n distinct primes.

%I A256493 #12 Dec 30 2019 14:13:41
%S A256493 1,3,19,171,1675,16683,166699,1666731,16666795,166666923,1666667179,
%T A256493 16666667691,166666668715,1666666670763,16666666674859,
%U A256493 166666666683051,1666666666699435,16666666666732203,166666666666797739,1666666666666928811,16666666666667190955
%N A256493 Number of factorizations of m^3 into at most 3 factors, where m is a product of exactly n distinct primes.
%C A256493 Also the number of n-partite partitions of (3)^n into at most 3 n-tuples.
%H A256493 Alois P. Heinz, <a href="/A256493/b256493.txt">Table of n, a(n) for n = 0..1000</a>
%H A256493 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13,-32,20).
%F A256493 G.f.: -(12*x^2-10*x+1)/((x-1)*(2*x-1)*(10*x-1)).
%F A256493 a(n) = (10^n + 3*2^n + 2)/6.
%e A256493 The a(1) = 3 factorizations of 2^3 into at most 3 factors are: 8, 2*4, 2*2*2.
%e A256493 The a(2) = 19 factorizations of (2*3)^3 into at most 3 factors are: 216, 2*108, 3*72, 4*54, 6*36, 8*27, 9*24, 12*18, 2*2*54, 2*3*36, 2*4*27, 2*6*18, 2*9*12, 3*3*24, 3*4*18, 3*6*12, 3*8*9, 4*6*9, 6*6*6.
%p A256493 a:= n-> (10^n + 3*2^n + 2)/6: seq(a(n), n=0..30);
%t A256493 LinearRecurrence[{13,-32,20},{1,3,19},30] (* _Harvey P. Dale_, Dec 30 2019 *)
%Y A256493 Row n=3 of A256384.
%K A256493 nonn,easy
%O A256493 0,2
%A A256493 _Alois P. Heinz_, Mar 30 2015