A177424 Exponent of the highest power of 2 dividing binomial(n^2,n).

0, 0, 1, 2, 2, 1, 4, 3, 3, 1, 3, 3, 2, 3, 5, 4, 4, 1, 3, 3, 5, 1, 4, 8, 3, 2, 3, 5, 6, 4, 6, 5, 5, 1, 3, 3, 5, 2, 6, 3, 3, 3, 4, 3, 4, 4, 5, 6, 4, 2, 3, 7, 5, 1, 5, 5, 3, 4, 6, 6, 7, 5, 7, 6, 6, 1, 3, 3, 5, 2, 6, 4, 7, 1, 3, 3, 3, 5, 6, 4, 4, 2, 6, 3, 5, 5, 4, 6, 6, 2, 4, 12, 6, 5, 6, 7, 5, 2, 3, 6, 5, 3, 6, 3, 6

Offset: 0

Author

Michel Lagneau, May 07 2010

Keywords

• nonn
• easy

Comments

a(n) is the largest integer such that 2^a(n) divides binomial(n^2,n)=A014062(n).

a(n) is the number of carries when adding n to n^2-n in base 2. - Robert Israel, Oct 23 2019

Examples

			For n = 6, binomial(36,6) = 1947792 = 2^4*3*7*11*17*31, the highest power of 2 is 2^4, and the exponent of 2^4 is a(6)=4.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000(n=0 to 1000 from Harvey P. Dale)
• Wikipedia, Kummer's theorem

Crossrefs

Cf. A014062, A000120, A177234

Programs

Maple

A007814 := proc(n) if type(n,'odd') then 0; else for p in ifactors(n)[2] do if op(1,p) = 2 then return op(2,p); end if; end do: end if; end proc:
A014062 := proc(n) binomial(n^2,n) ; end proc:
A177424 := proc(n) A007814(A014062(n)) ; end proc: seq(A177424(n),n=0..80) ;
# Alternative:
nc:= proc(a,b,c)
  local t;
  if c=0 and (a=0 or b=0) then return 0 fi;
  t:= (a mod 2) + (b mod 2) + c;
  if t < 2 then  procname(floor(a/2),floor(b/2),0)
  else  1 + procname(floor(a/2),floor(b/2),1)
  fi
end proc:
seq(nc(n,n^2-n,0),n=0..100); # Robert Israel, Oct 23 2019

Mathematica

IntegerExponent[Table[Binomial[n^2,n],{n,0,120}],2] (* Harvey P. Dale, Mar 31 2019 *)

PARI

valp(n,p=2)=my(s); while(n\=p, s+=n); s
a(n)=valp(n^2)-valp(n^2-n)-valp(n) \\ Charles R Greathouse IV, Jul 08 2022

Python

from math import comb
def A177424(n): return (~(m:=comb(n**2,n))& m-1).bit_length() # Chai Wah Wu, Jul 08 2022

Formula

a(n) = A007814(A014062(n)).

Extensions

Maple program replaced by a structured general version - R. J. Mathar, May 10 2010