A324466 Exponent of highest power of 2 that divides multinomial(3*n;n,n,n).

0, 1, 1, 4, 1, 2, 4, 6, 1, 2, 2, 7, 4, 5, 6, 8, 1, 2, 2, 5, 2, 3, 7, 9, 4, 5, 5, 9, 6, 7, 8, 10, 1, 2, 2, 5, 2, 3, 5, 7, 2, 3, 3, 10, 7, 8, 9, 11, 4, 5, 5, 8, 5, 6, 9, 11, 6, 7, 7, 11, 8, 9, 10, 12, 1, 2, 2, 5, 2, 3, 5, 7, 2, 3, 3, 8, 5, 6, 7, 9, 2, 3, 3, 6, 3, 4, 10, 12, 7, 8, 8

Offset: 0

Author

N. J. A. Sloane, Mar 02 2019

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000120 (analog for binomial coefficients), A006480, A007814, A324467.

Programs

Magma

[Valuation(Factorial(3*n)/Factorial(n)^3, 2): n in [0..100]]; // Vincenzo Librandi, Mar 10 2019

Maple

[seq(padic[ordp](combinat:-multinomial(3*n,n,n,n),2),n=0..128)];
# alternative:
f:= proc(n) local r,t;
  t:= 0; r:= 3*n;
  while r > 1 do t:= t + floor(r) - 3*floor(r/3); r:= r/2; od;
  t
end proc:
map(f, [$0..200]); # Robert Israel, Mar 03 2019

Mathematica

Table[IntegerExponent[(3 n)!/n!^3, 2], {n, 0, 100}] (* Vincenzo Librandi Mar 10 2019 *)

PARI

a(n) = valuation((3*n)!/n!^3, 2); \\ Michel Marcus, Mar 04 2019

Python

from math import factorial
def A324466(n): return (~(m:=factorial(3*n)//factorial(n)**3)& m-1).bit_length() # Chai Wah Wu, Jul 07 2022

Formula

a(2*n) = a(n). - Robert Israel, Mar 04 2019

From Amiram Eldar, Feb 21 2021: (Start)

a(n) = A007814(A006480(n)).

a(n) = 3*A000120(n) - A000120(3*n). (End)