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

%I A254609 #23 Mar 04 2023 04:27:52
%S A254609 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,
%T A254609 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,
%U A254609 5,5,5,1,1,5,5,5,1,1,1,1,1,5,5,1,1,1,5,5
%N A254609 Triangle read by rows: T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).
%C A254609 These are the generalized binomial coefficients associated with A060904.
%C A254609 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.
%C A254609 If T(n,k) != 0 mod 5, then n dominates k in base 5.
%C A254609 A194459(n) = number of ones in row n. - _Reinhard Zumkeller_, Feb 04 2015
%H A254609 Reinhard Zumkeller, <a href="/A254609/b254609.txt">Rows n = 0..124 of triangle, flattened</a>
%H A254609 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 A254609 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 A254609 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 A254609 T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).
%F A254609 T(n,k) = Product_{i=1..n} A060904(i)/(Product_{i=1..k} A060904(i)*Product_{i=1..n-k} A060904(i)).
%F A254609 T(n,k) = A060904(n)/n*(k/A060904(k)*T(n-1,k-1)+(n-k)/A060904(n-k)*T(n-1,k)).
%e A254609 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.
%e A254609 The triangle begins:
%e A254609 1
%e A254609 1, 1
%e A254609 1, 1, 1
%e A254609 1, 1, 1, 1
%e A254609 1, 1, 1, 1, 1
%e A254609 1, 5, 5, 5, 5, 1
%e A254609 1, 1, 5, 5, 5, 1, 1
%e A254609 1, 1, 1, 5, 5, 1, 1, 1
%e A254609 1, 1, 1, 1, 5, 1, 1, 1, 1
%e A254609 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%e A254609 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1
%e A254609 1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1
%e A254609 1, 1, 1, 5, 5, 1, 1, 1, 5, 5, 1, 1, 1
%e A254609 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1
%e A254609 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%o A254609 P=[0]+[5^valuation(i,5) for i in [1..100]]
%o A254609 [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 A254609 (Haskell)
%o A254609 import Data.List (inits)
%o A254609 a254609 n k = a254609_tabl !! n !! k
%o A254609 a254609_row n = a254609_tabl !! n
%o A254609 a254609_tabl = zipWith (map . div)
%o A254609    a243757_list $ zipWith (zipWith (*)) xss $ map reverse xss
%o A254609    where xss = tail $ inits a243757_list
%o A254609 -- _Reinhard Zumkeller_, Feb 04 2015
%Y A254609 Cf. A060904, A243757, A234957, A242849, A082907.
%Y A254609 Cf. A194459.
%K A254609 nonn,tabl
%O A254609 0,17
%A A254609 _Tom Edgar_, Feb 02 2015