OFFSET
0,23
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.
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.
FORMULA
EXAMPLE
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
PROG
(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]
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tom Edgar, Feb 06 2015
STATUS
approved