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

%I A324466 #37 Jul 08 2022 08:22:51
%S A324466 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,
%T A324466 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,
%U A324466 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
%N A324466 Exponent of highest power of 2 that divides multinomial(3*n;n,n,n).
%H A324466 Robert Israel, <a href="/A324466/b324466.txt">Table of n, a(n) for n = 0..10000</a>
%F A324466 a(2*n) = a(n). - _Robert Israel_, Mar 04 2019
%F A324466 From _Amiram Eldar_, Feb 21 2021: (Start)
%F A324466 a(n) = A007814(A006480(n)).
%F A324466 a(n) = 3*A000120(n) - A000120(3*n). (End)
%p A324466 [seq(padic[ordp](combinat:-multinomial(3*n,n,n,n),2),n=0..128)];
%p A324466 # alternative:
%p A324466 f:= proc(n) local r,t;
%p A324466   t:= 0; r:= 3*n;
%p A324466   while r > 1 do t:= t + floor(r) - 3*floor(r/3); r:= r/2; od;
%p A324466   t
%p A324466 end proc:
%p A324466 map(f, [$0..200]); # _Robert Israel_, Mar 03 2019
%t A324466 Table[IntegerExponent[(3 n)!/n!^3, 2], {n, 0, 100}] (* _Vincenzo Librandi_ Mar 10 2019 *)
%o A324466 (PARI) a(n) = valuation((3*n)!/n!^3, 2); \\ _Michel Marcus_, Mar 04 2019
%o A324466 (Magma) [Valuation(Factorial(3*n)/Factorial(n)^3, 2): n in [0..100]]; // _Vincenzo Librandi_, Mar 10 2019
%o A324466 (Python)
%o A324466 from math import factorial
%o A324466 def A324466(n): return (~(m:=factorial(3*n)//factorial(n)**3)& m-1).bit_length() # _Chai Wah Wu_, Jul 07 2022
%Y A324466 Cf. A000120 (analog for binomial coefficients), A006480, A007814, A324467.
%K A324466 nonn,look
%O A324466 0,4
%A A324466 _N. J. A. Sloane_, Mar 02 2019