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A243756 Triangle read by rows: T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).

%I A243756 #24 Mar 04 2023 04:27:32
%S A243756 1,1,1,1,1,1,1,1,1,1,1,4,4,4,1,1,1,4,4,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,
%T A243756 1,1,1,4,4,4,1,4,4,4,1,1,1,4,4,1,1,4,4,1,1,1,1,1,4,1,1,1,4,1,1,1,1,1,
%U A243756 1,1,1,1,1,1,1,1,1,1,1,4,4,4,1,4,4,4,1,4,4,4,1
%N A243756 Triangle read by rows: T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).
%C A243756 The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 4 using the traditional addition algorithm.
%C A243756 If T(n,k) != 0 mod 4, then n dominates k in base 4.
%H A243756 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 A243756 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 A243756 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 A243756 T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).
%F A243756 T(n,k) = Product_{i=1..n} A234957(i)/(Product_{i=1..k} A234957(i)*Product_{i=1..n-k} A234957(i)).
%F A243756 T(n,k) = A234957(n)/n*(k/A234957(k)*T(n-1,k-1)+(n-k)/A234957(n-k)*T(n-1,k)).
%e A243756 The triangle begins:
%e A243756 1;
%e A243756 1, 1;
%e A243756 1, 1, 1;
%e A243756 1, 1, 1, 1;
%e A243756 1, 4, 4, 4, 1;
%e A243756 1, 1, 4, 4, 1, 1;
%e A243756 1, 1, 1, 4, 1, 1, 1;
%e A243756 1, 1, 1, 1, 1, 1, 1, 1;
%e A243756 1, 4, 4, 4, 1, 4, 4, 4, 1;
%e A243756 1, 1, 4, 4, 1, 1, 4, 4, 1, 1;
%e A243756 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1;
%e A243756 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%o A243756 (Sage)
%o A243756 m=50
%o A243756 T=[0]+[4^valuation(i, 4) for i in [1..m]]
%o A243756 Table=[[prod(T[1:i+1])/(prod(T[1:j+1])*prod(T[1:i-j+1])) for j in [0..i]] for i in [0..m-1]]
%o A243756 [x for sublist in Table for x in sublist]
%Y A243756 Cf. A082907, A234957, A242849, A242954.
%K A243756 nonn,tabl
%O A243756 0,12
%A A243756 _Tom Edgar_, Jun 09 2014