A242849 - OEIS

%I #22 Mar 04 2023 04:27:26

%S 1,1,1,1,1,1,1,3,3,1,1,1,3,1,1,1,1,1,1,1,1,1,3,3,1,3,3,1,1,1,3,1,1,3,

%T 1,1,1,1,1,1,1,1,1,1,1,1,9,9,3,9,9,3,9,9,1,1,1,9,3,3,9,3,3,9,1,1,1,1,

%U 1,3,3,3,3,3,3,1,1,1,1,3,3,1,9,9,3,9,9,1

%N Triangle read by rows: T(n,k) = A060828(n)/(A060828(k) * A060828(n-k)).

%C This triangle can be obtained by replacing each entry of Pascal's Triangle by the largest power of 3 dividing that entry.

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

%C If T(n,k) != 0 mod 3, then n dominates k in base 3.

%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.

%H Tyler Ball and Daniel Juda, <a href="https://scholar.rose-hulman.edu/rhumj/vol14/iss2/2">Dominance over N</a>, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

%H E. Burlachenko, <a href="https://arxiv.org/abs/1612.00970">Fractal generalized Pascal matrices</a>, arXiv:1612.00970 [math.NT], 2016. See p. 7.

%H Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

%F T(n,k) = A060828(n)/(A060828(k) * A060828(n-k)).

%F T(n,k) = Product_{i=1..n} A038500(i)/(Product_{i=1..k} A038500(i)*Product_{i=1..n-k} A038500(i)).

%F T(n,k) = A038500(n)/n*(k/A038500(k)*T(n-1,k-1)+(n-k)/A038500(n-k)*T(n-1,k)).

%e The triangle begins

%e 1

%e 1 1

%e 1 1 1

%e 1 3 3 1

%e 1 1 3 1 1

%e 1 1 1 1 1 1

%e 1 3 3 1 3 3 1.

%t s3[n_] := 3^IntegerExponent[n!, 3];

%t T[n_, k_] := s3[n]/(s3[k] s3[n-k]);

%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 06 2018 *)

%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 [x for sublist in Table for x in sublist]

%Y Cf. A007318, A038500, A060828, A082907.

%K nonn,tabl

%O 0,8

%A _Tom Edgar_, May 23 2014