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

0,47

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.

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.

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Row sums: A249343.

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

Sequence in context: A365096 A102297 A269570 * A098398 A306211 A131714

Adjacent sequences: A243756 A243757 A243758 * A243760 A243761 A243762

Keyword

nonn,tabl

Author

Tom Edgar, Jun 10 2014

Extensions

Name clarified by Antti Karttunen, Oct 28 2014

Status

approved