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%I #22 Mar 04 2023 04:27:49
%S 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,
%T 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,
%U 1,1,1,1,1,1,1,1,1,1,1,6,6,6,6,6,1,6,6,6
%N Triangle read by rows: T(n,k) = A243758(n)/(A243758(k)*A243758(n-k)).
%C These are the generalized binomial coefficients associated with A234959.
%C 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.
%C If T(n,k) != 0 mod 6, then n dominates k in base 6.
%H Reinhard Zumkeller, <a href="/A254730/b254730.txt">Rows n = 0..125 of triangle, flattened</a>
%H Tyler Ball, Tom Edgar, and Daniel Juda, <a href="http://dx.doi.org/10.4169/math.mag.87.2.135">Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem</a>, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
%H Tyler Ball and Daniel Juda, <a href="https://scholar.rose-hulman.edu/rhumj/vol14/iss2/2">Dominance over N</a>, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
%H Tom Edgar and Michael Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Edgar/edgar3.html">Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
%F T(n,k) = A243758(n)/(A243758(k)*A243758(n-k)).
%F T(n,k) = Product_{i=1..n} A234959(i)/(Product_{i=1..k} A234959(i)*Product_{i=1..n-k} A234959(i)).
%F T(n,k) = A234959(n)/n*(k/A234959(k)*T(n-1,k-1)+(n-k)/A234959(n-k)*T(n-1,k)).
%e 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.
%e The triangle begins:
%e 1
%e 1, 1
%e 1, 1, 1
%e 1, 1, 1, 1
%e 1, 1, 1, 1, 1
%e 1, 1, 1, 1, 1, 1
%e 1, 6, 6, 6, 6, 6, 1
%e 1, 1, 6, 6, 6, 6, 1, 1
%e 1, 1, 1, 6, 6, 6, 1, 1, 1
%e 1, 1, 1, 1, 6, 6, 1, 1, 1, 1
%e 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%e 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1
%e 1, 1, 6, 6, 6, 6, 1, 1, 6, 6, 6, 6, 1, 1
%e 1, 1, 1, 6, 6, 6, 1, 1, 1, 6, 6, 6, 1, 1, 1
%e 1, 1, 1, 1, 6, 6, 1, 1, 1, 1, 6, 6, 1, 1, 1, 1
%e 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%o (Sage)
%o P=[0]+[6^valuation(i,6) for i in [1..100]]
%o [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]
%o (Haskell)
%o import Data.List (inits)
%o a254730 n k = a254730_tabl !! n !! k
%o a254730_row n = a254730_tabl !! n
%o a254730_tabl = zipWith (map . div)
%o a243758_list $ zipWith (zipWith (*)) xss $ map reverse xss
%o where xss = tail $ inits a243758_list
%o -- _Reinhard Zumkeller_, Feb 09 2015
%Y Cf. A234959, A243758, A242849, A082907, A254609.
%K nonn,tabl
%O 0,23
%A _Tom Edgar_, Feb 06 2015
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