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A238453 Triangle read by rows: T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).

%I A238453 #47 Nov 14 2018 15:27:15
%S A238453 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,4,8,8,4,1,1,2,8,8,8,2,1,1,6,12,24,24,
%T A238453 12,6,1,1,4,24,24,48,24,24,4,1,1,6,24,72,72,72,72,24,6,1,1,4,24,48,
%U A238453 144,72,144,48,24,4,1,1,10,40,120,240,360,360,240,120
%N A238453 Triangle read by rows: T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).
%C A238453 We assume that A001088(0)=1 since it would be the empty product.
%C A238453 These are the generalized binomial coefficients associated with Euler's totient function A000010.
%C A238453 Another name might be the totienomial coefficients.
%H A238453 Reinhard Zumkeller, <a href="/A238453/b238453.txt">Rows n = 0..125 of triangle, flattened</a>
%H A238453 Tom Edgar, <a href="http://www.emis.de/journals/INTEGERS/papers/o62/o62.Abstract.html">Totienomial Coefficients</a>, INTEGERS, 14 (2014), #A62.
%H A238453 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 A238453 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 A238453 T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).
%F A238453 T(n,k) = prod_{i=1..n} A000010(i)/(prod_{i=1..k} A000010(i)*prod_{i=1..n-k} A000010(i)).
%F A238453 T(n,k) = A000010(n)/n*(k/A000010(k)*T(n-1,k-1)+(n-k)/A000010(n-k)*T(n-1,k)).
%F A238453 T(n+1, 2) = A083542(n). - _Michael Somos_, Aug 26 2014
%F A238453 T(n,k) = Product_{i=1..k} (phi(n+1-i)/phi(i)), where phi is Euler's totient function (A000010). - _Werner Schulte_, Nov 14 2018
%e A238453 The first five terms in Euler's totient function are 1,1,2,2,4 and so T(4,2) = 2*2*1*1/((1*1)*(1*1))=4 and T(5,3) = 4*2*2*1*1/((2*1*1)*(1*1))=8.
%e A238453 The triangle begins
%e A238453 1
%e A238453 1 1
%e A238453 1 1 1
%e A238453 1 2 2 1
%e A238453 1 2 4 2 1
%e A238453 1 4 8 8 4 1
%e A238453 1 2 8 8 8 2 1
%t A238453 f[n_] := Product[EulerPhi@ k, {k, n}]; Table[f[n]/(f[k] f[n - k]), {n, 0, 11}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 19 2016 *)
%o A238453 (Sage)
%o A238453 q=100 #change q for more rows
%o A238453 P=[euler_phi(i) for i in [0..q]]
%o A238453 [[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.
%o A238453 (Haskell)
%o A238453 a238453 n k = a238453_tabl !! n !! k
%o A238453 a238453_row n = a238453_tabl !! n
%o A238453 a238453_tabl = [1] : f [1] a000010_list where
%o A238453    f xs (z:zs) = (map (div y) $ zipWith (*) ys $ reverse ys) : f ys zs
%o A238453      where ys = y : xs; y = head xs * z
%o A238453 -- _Reinhard Zumkeller_, Feb 27 2014
%o A238453 (PARI) T(n,k)={prod(i=1, k, eulerphi(n+1-i)/eulerphi(i))} \\ _Andrew Howroyd_, Nov 13 2018
%Y A238453 Cf. A000010, A001088, A083542.
%K A238453 nonn,tabl
%O A238453 0,8
%A A238453 _Tom Edgar_, Feb 26 2014