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A255914 Triangle read by rows: T(n,k) = A007318(n,k)*A238453(n,k).

%I A255914 #12 Jan 25 2016 14:23:38
%S A255914 1,1,1,1,2,1,1,6,6,1,1,8,24,8,1,1,20,80,80,20,1,1,12,120,160,120,12,1,
%T A255914 1,42,252,840,840,252,42,1,1,32,672,1344,3360,1344,672,32,1,1,54,864,
%U A255914 6048,9072,9072,6048,864,54,1,1,40,1080,5760,30240,18144,30240
%N A255914 Triangle read by rows: T(n,k) = A007318(n,k)*A238453(n,k).
%C A255914 These are the generalized binomial coefficients associated with the sequence A002618.
%H A255914 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 A255914 T(n,k) = Product_{i=1..n} A002618(i)/(Product_{i=1..k} A002618(i)*Product_{i=1..n-k} A002618(i)).
%F A255914 T(n,k) = A002618(n)/n*(k/A002618(k)*T(n-1,k-1)+(n-k)/A002618(n-k)*T(n-1,k)).
%e A255914 The first five terms in A002618 (n*phi(n)) are 1, 2, 6, 8, 20 and so T(4,2) = 8*6*2*1/((2*1)*(2*1)) = 24 and T(5,3) = 20*8*6*2*1/((6*2*1)*(2*1)) = 80.
%e A255914 The triangle begins:
%e A255914 1;
%e A255914 1, 1;
%e A255914 1, 2, 1;
%e A255914 1, 6, 6, 1;
%e A255914 1, 8, 24, 8, 1;
%e A255914 1, 20, 80, 80, 20, 1;
%e A255914 1, 12, 120, 160, 120, 12, 1;
%e A255914 1, 42, 252, 840, 840, 252, 42, 1
%o A255914 (Sage)
%o A255914 q=100 #change q for more rows
%o A255914 P=[i*euler_phi(i) for i in [0..q]]
%o A255914 [[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 A255914 Cf. A002618, A238453, A007318.
%K A255914 nonn,tabl
%O A255914 0,5
%A A255914 _Tom Edgar_, Mar 10 2015