A243759 Triangle T(m,k): exponent of the highest power of 3 dividing the binomial coefficient binomial(m,k).

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2, 2, 1, 2, 2, 0, 0, 0, 2, 1, 1, 2, 1, 1, 2, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 2, 1, 2, 2

Offset: 0

Author

Tom Edgar, Jun 10 2014

Keywords

• nonn
• tabl

Comments

T(m,k) is the number of 'carries' that occur when adding k and n-k in base 3 using the traditional addition algorithm.

Examples

			The triangle begins:
0,
0, 0,
0, 0, 0,
0, 1, 1, 0;
0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0;
0, 1, 1, 0, 1, 1, 0;
0, 0, 1, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0;
0, 2, 2, 1, 2, 2, 1, 2, 2, 0;
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10010
• Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Crossrefs

Row sums: A249343.

Cf. A007318, A007949, A083093, A065040, A242849, A060828, A038500.

Programs

Maple

A243759:= (m,k) -> padic[ordp](binomial(m,k),3);
for m from 0 to 50 do
  seq(A243759(m,k),k=0..m)
od;   # Robert Israel, Jun 15 2014

Mathematica

T[m_, k_] := IntegerExponent[Binomial[m, k], 3];
Table[T[m, k], {m, 0, 12}, {k, 0, m}] // Flatten (* Jean-François Alcover, Jun 05 2022 *)

Sage

m=50
T=[0]+[3^valuation(i, 3) for i in [1..m]]
Table=[[prod(T[1:i+1])/(prod(T[1:j+1])*prod(T[1:i-j+1])) for j in [0..i]] for i in [0..m-1]]
[log(Integer(x),base=3) for sublist in Table for x in sublist]

Scheme

(define (A243759 n) (A007949 (A007318 n))) ;; Antti Karttunen, Oct 28 2014

Formula

T(m,k) = log_3(A242849(m,k)).

From Antti Karttunen, Oct 28 2014: (Start)

a(n) = A007949(A007318(n)).

a(n) * A083093(n) = 0 and a(n) + A083093(n) > 0 for all n.

(End)

Extensions

Name clarified by Antti Karttunen, Oct 28 2014