A242849 Triangle read by rows: T(n,k) = A060828(n)/(A060828(k) * A060828(n-k)).

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, 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, 1, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 1, 9, 9, 3, 9, 9, 1

Offset: 0

Author

Tom Edgar, May 23 2014

Keywords

• nonn
• tabl

Comments

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

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.

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

Examples

			The triangle begins
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.
		

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.
• Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
• E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 7.
• Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Crossrefs

Cf. A007318, A038500, A060828, A082907.

Programs

Mathematica

s3[n_] := 3^IntegerExponent[n!, 3];
T[n_, k_] := s3[n]/(s3[k] s3[n-k]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

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

Formula

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

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

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