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

%I A243759 #28 Jun 05 2022 03:39:31
%S A243759 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 A243759 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 A243759 0,1,1,1,1,1,1,0,0,0,0,1,1,0,2,2,1,2,2
%N A243759 Triangle T(m,k): exponent of the highest power of 3 dividing the binomial coefficient binomial(m,k).
%C A243759 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 A243759 Robert Israel, <a href="/A243759/b243759.txt">Table of n, a(n) for n = 0..10010</a>
%H A243759 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 A243759 T(m,k) = log_3(A242849(m,k)).
%F A243759 From _Antti Karttunen_, Oct 28 2014: (Start)
%F A243759 a(n) = A007949(A007318(n)).
%F A243759 a(n) * A083093(n) = 0 and a(n) + A083093(n) > 0 for all n.
%F A243759 (End)
%e A243759 The triangle begins:
%e A243759 0,
%e A243759 0, 0,
%e A243759 0, 0, 0,
%e A243759 0, 1, 1, 0;
%e A243759 0, 0, 1, 0, 0;
%e A243759 0, 0, 0, 0, 0, 0;
%e A243759 0, 1, 1, 0, 1, 1, 0;
%e A243759 0, 0, 1, 0, 0, 1, 0, 0;
%e A243759 0, 0, 0, 0, 0, 0, 0, 0, 0;
%e A243759 0, 2, 2, 1, 2, 2, 1, 2, 2, 0;
%p A243759 A243759:= (m,k) -> padic[ordp](binomial(m,k),3);
%p A243759 for m from 0 to 50 do
%p A243759   seq(A243759(m,k),k=0..m)
%p A243759 od;   # _Robert Israel_, Jun 15 2014
%t A243759 T[m_, k_] := IntegerExponent[Binomial[m, k], 3];
%t A243759 Table[T[m, k], {m, 0, 12}, {k, 0, m}] // Flatten (* _Jean-François Alcover_, Jun 05 2022 *)
%o A243759 (Sage)
%o A243759 m=50
%o A243759 T=[0]+[3^valuation(i, 3) for i in [1..m]]
%o A243759 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 A243759 [log(Integer(x),base=3) for sublist in Table for x in sublist]
%o A243759 (Scheme) (define (A243759 n) (A007949 (A007318 n))) ;; _Antti Karttunen_, Oct 28 2014
%Y A243759 Row sums: A249343.
%Y A243759 Cf. A007318, A007949, A083093, A065040, A242849, A060828, A038500.
%K A243759 nonn,tabl
%O A243759 0,47
%A A243759 _Tom Edgar_, Jun 10 2014
%E A243759 Name clarified by _Antti Karttunen_, Oct 28 2014