A254609 Triangle read by rows: T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).

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

Reinhard Zumkeller, Rows n = 0..124 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

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

Cf. A060904, A243757, A234957, A242849, A082907.

Cf. A194459.

Sequence in context: A193887 A196998 A232614 * A133707 A171372 A083945

Adjacent sequences: A254606 A254607 A254608 * A254610 A254611 A254612

Keyword

nonn,tabl

Author

Tom Edgar, Feb 02 2015

Status

approved