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

0,8

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.

Links

Table of n, a(n) for n=0..87.

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.

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)).

Example

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.

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 *)

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

Crossrefs

Cf. A007318, A038500, A060828, A082907.

Sequence in context: A114231 A079075 A086703 * A072917 A319861 A114266

Adjacent sequences: A242846 A242847 A242848 * A242850 A242851 A242852

Keyword

nonn,tabl

Author

Tom Edgar, May 23 2014

Status

approved