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A246465 Triangle read by rows: T(n,k) = A085056(n)/(A085056(k) * A085056(n-k)).

%I A246465 #16 Jan 25 2016 14:23:16
%S A246465 1,1,1,1,1,1,1,1,1,1,1,2,2,2,1,1,1,2,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,
%T A246465 1,1,1,4,4,4,2,4,4,4,1,1,3,12,12,6,6,12,12,3,1,1,1,3,12,6,6,6,12,3,1,
%U A246465 1,1,1,1,3,6,6,6,6,3,1,1,1,1,2,2,2,3,12,12
%N A246465 Triangle read by rows: T(n,k) = A085056(n)/(A085056(k) * A085056(n-k)).
%C A246465 We assume that A085056(0)=1 since it would be the empty product.
%C A246465 These are the generalized binomial coefficients associated with the sequence A003557.
%H A246465 Tom Edgar, <a href="http://www.emis.de/journals/INTEGERS/papers/o62/o62.Abstract.html">Totienomial Coefficients</a>, INTEGERS, 14 (2014), #A62.
%H A246465 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.
%H A246465 Donald E. Knuth and Herbert S. Wilf, <a href="http://www.math.upenn.edu/~wilf/website/dm36.pdf">The power of a prime that divides a generalized binomial coefficient</a>, J. Reine Angew. Math., 396:212-219, 1989.
%F A246465 T(n,k) = A085056(n)/(A085056(k) * A085056(n-k)).
%F A246465 T(n,k) = prod_{i=1..n} A003557(i)/(prod_{i=1..k} A003557(i)*prod_{i=1..n-k} A003557(i)).
%F A246465 T(n,k) = A003557(n)/n*(k/A003557(k)*T(n-1,k-1)+(n-k)/A003557(n-k)*T(n-1,k)).
%e A246465 The first five terms in A003557 are: 1, 1, 1, 2, 1 and so T(4,2) = 2*1*1*1/((1*1)*(1*1))=2 and T(5,4) = 1*2*1*1*1/((2*1*1*1)*(1))=1.
%e A246465 The triangle begins:
%e A246465 1,
%e A246465 1, 1,
%e A246465 1, 1, 1,
%e A246465 1, 1, 1, 1,
%e A246465 1, 2, 2, 2, 1,
%e A246465 1, 1, 2, 2, 1, 1,
%e A246465 1, 1, 1, 2, 1, 1, 1,
%e A246465 1, 1, 1, 1, 1, 1, 1, 1,
%e A246465 1, 4, 4, 4, 2, 4, 4, 4, 1,
%e A246465 1, 3, 12, 12, 6, 6, 12, 12, 3, 1.
%o A246465 (Sage)
%o A246465 q=100 #change q for more rows
%o A246465 P=[0]+[n/prod([x for x in prime_divisors(n)]) for n in [1..q]]
%o A246465 [[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] # generates the triangle up to q rows.
%Y A246465 Cf. A003557, A085056, A048804.
%K A246465 nonn,tabl
%O A246465 0,12
%A A246465 _Tom Edgar_, Aug 27 2014