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

0,3

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

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

Example

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.

Crossrefs

Row n=4 of A257463.

Sequence in context: A228462 A217641 A213574 * A119783 A199042 A362305

Adjacent sequences: A253256 A253257 A253258 * A253260 A253261 A253262

Keyword

nonn,easy

Author

Alois P. Heinz, Apr 30 2015

Status

approved