A243758 -id:A243758 - cp's OEIS frontend

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

Tom Edgar, Feb 06 2015

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.

			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

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.

Cf. A234959, A243758, A242849, A082907, A254609.

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

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]

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)).

A234959 Highest power of 6 dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6

Offset: 1

Tom Edgar, Jan 01 2014

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 6.

			Since 12 = 6 * 2, a(12) = 6. Likewise, since 6 does not divide 13, a(13) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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.

Cf. A006519, A038500, A122841, A234957, A243758.

a234959 = f 1 where
   f y x = if m == 0 then f (y * 6) x' else y  where (x', m) = divMod x 6
-- Reinhard Zumkeller, Feb 09 2015

6^Table[IntegerExponent[n, 6], {n, 84}] (* Alonso del Arte, Jan 01 2014 *)

a(n)=6^valuation(n,6) \\ Charles R Greathouse IV, Aug 05 2015

n=200 #change n for more terms
[6^(valuation(i,6)) for i in [1..n]]

a(n) = 6^(valuation(n,6)).
a(n) = 6^A122841(n). - Joerg Arndt, Jan 02 2014
G.f.: x/(1 - x) + 5 * Sum_{k>=1} 6^(k-1)*x^(6^k)/(1 - x^(6^k)). - Ilya Gutkovskiy, Jul 10 2019

cp's OEIS Frontend

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

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