A253259 Number of factorizations of m^n into 4 factors, where m is a product of exactly 4 distinct primes and each factor is a product of n primes (counted with multiplicity).
1, 1, 17, 93, 465, 1746, 5741, 16238, 41650, 97407, 212412, 434767, 845366, 1569344, 2801696, 4828140, 8069053, 13114785, 20796651, 32242621, 48986553, 73052382, 107114645, 154621230, 220021932, 308940815, 428492880, 587520315, 797019526, 1070458096
Offset: 0
Examples
a(2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*4.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. This sequence is column 4 of table f(m,n) on page 40.
- Math StackExchange, Number of ways to partition 40 balls with 4 colors into 4 baskets
- Marko Riedel, Maple program to compute sequence from cycle indices
Crossrefs
Row n=4 of A257463.
Formula
[A^n B^n C^n D^n] Z(S_4)(Z(S_n)(A+B+C+D)) with Z(S_q) the cycle index of the symmetric group; parenthesis denote the canonical substitution of the argument into the cycle index. - Marko Riedel, Feb 06 2016
G.f.: (x^18 +6*x^17 +58*x^16 +213*x^15 +646*x^14 +1415*x^13 +2515*x^12 +3554*x^11 +4296*x^10 +4248*x^9 +3578*x^8 +2452*x^7 +1421*x^6 +628*x^5 +240*x^4 +61*x^3 +12*x^2-x+1) /((1-x)^10 *(1+x)^5 *(1+x+x^2)^3 *(1+x^2)). [This was found by Will Orrick and confirmed by Marko Riedel, see the StackExchange link above.] - Alois P. Heinz, Feb 09 2016