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A254609
Triangle read by rows: T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).
3
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 5, 5
OFFSET
0,17
COMMENTS
These are the generalized binomial coefficients associated with A060904.
The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 5 using the traditional addition algorithm.
If T(n,k) != 0 mod 5, then n dominates k in base 5.
A194459(n) = number of ones in row n. - Reinhard Zumkeller, Feb 04 2015
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.
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) = A243757(n)/(A243757(k)*A243757(n-k)).
T(n,k) = Product_{i=1..n} A060904(i)/(Product_{i=1..k} A060904(i)*Product_{i=1..n-k} A060904(i)).
T(n,k) = A060904(n)/n*(k/A060904(k)*T(n-1,k-1)+(n-k)/A060904(n-k)*T(n-1,k)).
EXAMPLE
The first five terms in A060904 are 1, 1, 1, 1, and 5 and so T(4,2) = 1*1*1*1/((1*1)*(1*1))=1 and T(5,3) = 5*1*1*1*1/((1*1*1)*(1*1))=5.
The triangle begins:
1
1, 1
1, 1, 1
1, 1, 1, 1
1, 1, 1, 1, 1
1, 5, 5, 5, 5, 1
1, 1, 5, 5, 5, 1, 1
1, 1, 1, 5, 5, 1, 1, 1
1, 1, 1, 1, 5, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1
1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1
1, 1, 1, 5, 5, 1, 1, 1, 5, 5, 1, 1, 1
1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
PROG
P=[0]+[5^valuation(i, 5) 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)
a254609 n k = a254609_tabl !! n !! k
a254609_row n = a254609_tabl !! n
a254609_tabl = zipWith (map . div)
a243757_list $ zipWith (zipWith (*)) xss $ map reverse xss
where xss = tail $ inits a243757_list
-- Reinhard Zumkeller, Feb 04 2015
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tom Edgar, Feb 02 2015
STATUS
approved