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).

1, 1, 5, 10, 23, 40, 73, 114, 180, 262, 379, 521, 712, 938, 1228, 1567, 1986, 2469, 3052, 3715, 4499, 5383, 6410, 7558, 8875, 10335, 11991, 13816, 15865, 18110, 20611, 23336, 26350, 29620, 33213, 37095, 41338, 45904, 50870, 56197, 61964, 68131, 74782, 81873

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.

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

Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Formula

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).

Example

a(2) = 5: (2*3*5)^2 = 900 = 10*10*9 = 15*10*6 = 15*15*4 = 25*6*6 = 25*9*4.
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.

Maple

a:= n-> coeff(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, n+1), x, n):
seq(a(n), n=0..60);

Mathematica

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 *)
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 *)

Crossrefs

Row n=3 of A257463.

Sequence in context: A341638 A280002 A260567 * A295731 A175661 A355552

Adjacent sequences: A257461 A257462 A257463 * A257465 A257466 A257467

Keyword

nonn,easy

Author

Alois P. Heinz, Apr 24 2015

Status

approved