A254730 Triangle read by rows: T(n,k) = A243758(n)/(A243758(k)*A243758(n-k)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 6, 1, 1, 1, 6, 6, 6, 6, 1, 1, 1, 1, 1, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6

Offset: 0

Author

Tom Edgar, Feb 06 2015

Keywords

• nonn
• tabl

Comments

These are the generalized binomial coefficients associated with A234959.

The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 6 using the traditional addition algorithm.

If T(n,k) != 0 mod 6, then n dominates k in base 6.

Examples

			The first six terms in A234959 are 1, 1, 1, 1, 1 and 6 and so T(4,2) = 1*1*1*1/((1*1)*(1*1))=1 and T(6,3) = 6*1*1*1*1*1/((1*1*1)*(1*1*1))=6.
The triangle begins:
1
1, 1
1, 1, 1
1, 1, 1, 1
1, 1, 1, 1, 1
1, 1, 1, 1, 1, 1
1, 6, 6, 6, 6, 6, 1
1, 1, 6, 6, 6, 6, 1, 1
1, 1, 1, 6, 6, 6, 1, 1, 1
1, 1, 1, 1, 6, 6, 1, 1, 1, 1
1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1
1, 1, 6, 6, 6, 6, 1, 1, 6, 6, 6, 6, 1, 1
1, 1, 1, 6, 6, 6, 1, 1, 1, 6, 6, 6, 1, 1, 1
1, 1, 1, 1, 6, 6, 1, 1, 1, 1, 6, 6, 1, 1, 1, 1
1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• 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.
• 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. A234959, A243758, A242849, A082907, A254609.

Programs

Haskell

import Data.List (inits)
a254730 n k = a254730_tabl !! n !! k
a254730_row n = a254730_tabl !! n
a254730_tabl = zipWith (map . div)
   a243758_list $ zipWith (zipWith (*)) xss $ map reverse xss
   where xss = tail $ inits a243758_list
-- Reinhard Zumkeller, Feb 09 2015

Sage

P=[0]+[6^valuation(i,6) for i in [1..100]]
[m for sublist in [[mul(P[1:n+1])/(mul(P[1:k+1])*mul(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] for m in sublist]

Formula

T(n,k) = A243758(n)/(A243758(k)*A243758(n-k)).

T(n,k) = Product_{i=1..n} A234959(i)/(Product_{i=1..k} A234959(i)*Product_{i=1..n-k} A234959(i)).

T(n,k) = A234959(n)/n*(k/A234959(k)*T(n-1,k-1)+(n-k)/A234959(n-k)*T(n-1,k)).