A243759 - OEIS

%I #28 Jun 05 2022 03:39:31

%S 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,

%T 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,

%U 0,1,1,1,1,1,1,0,0,0,0,1,1,0,2,2,1,2,2

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

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

%H Robert Israel, <a href="/A243759/b243759.txt">Table of n, a(n) for n = 0..10010</a>

%H Tyler Ball, Tom Edgar, and Daniel Juda, <a href="http://dx.doi.org/10.4169/math.mag.87.2.135">Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem</a>, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

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

%F From _Antti Karttunen_, Oct 28 2014: (Start)

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

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

%F (End)

%e The triangle begins:

%e 0,

%e 0, 0,

%e 0, 0, 0,

%e 0, 1, 1, 0;

%e 0, 0, 1, 0, 0;

%e 0, 0, 0, 0, 0, 0;

%e 0, 1, 1, 0, 1, 1, 0;

%e 0, 0, 1, 0, 0, 1, 0, 0;

%e 0, 0, 0, 0, 0, 0, 0, 0, 0;

%e 0, 2, 2, 1, 2, 2, 1, 2, 2, 0;

%p A243759:= (m,k) -> padic[ordp](binomial(m,k),3);

%p for m from 0 to 50 do

%p   seq(A243759(m,k),k=0..m)

%p od;   # _Robert Israel_, Jun 15 2014

%t T[m_, k_] := IntegerExponent[Binomial[m, k], 3];

%t Table[T[m, k], {m, 0, 12}, {k, 0, m}] // Flatten (* _Jean-François Alcover_, Jun 05 2022 *)

%o (Sage)

%o m=50

%o T=[0]+[3^valuation(i, 3) for i in [1..m]]

%o 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]]

%o [log(Integer(x),base=3) for sublist in Table for x in sublist]

%o (Scheme) (define (A243759 n) (A007949 (A007318 n))) ;; _Antti Karttunen_, Oct 28 2014

%Y Row sums: A249343.

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

%K nonn,tabl

%O 0,47

%A _Tom Edgar_, Jun 10 2014

%E Name clarified by _Antti Karttunen_, Oct 28 2014