A253259 - OEIS

%I #30 Feb 09 2016 05:56:57

%S 1,1,17,93,465,1746,5741,16238,41650,97407,212412,434767,845366,

%T 1569344,2801696,4828140,8069053,13114785,20796651,32242621,48986553,

%U 73052382,107114645,154621230,220021932,308940815,428492880,587520315,797019526,1070458096

%N 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).

%H Alois P. Heinz, <a href="/A253259/b253259.txt">Table of n, a(n) for n = 0..1000</a>

%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 4 of table f(m,n) on page 40.

%H Math StackExchange, <a href="http://math.stackexchange.com/questions/1641433/">Number of ways to partition 40 balls with 4 colors into 4 baskets</a>

%H Marko Riedel, <a href="/A253259/a253259.maple.txt">Maple program to compute sequence from cycle indices</a>

%F [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

%F 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

%e 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.

%Y Row n=4 of A257463.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Apr 30 2015