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A243759
Triangle T(m,k): exponent of the highest power of 3 dividing the binomial coefficient binomial(m,k).
3
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
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
KEYWORD
nonn,tabl
AUTHOR
Tom Edgar, Jun 10 2014
EXTENSIONS
Name clarified by Antti Karttunen, Oct 28 2014
STATUS
approved