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A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)). 2


%S 1,1,1,1,15,1,1,80,80,1,1,240,1280,240,1,1,624,9984,9984,624,1,1,1200,

%T 49920,149760,49920,1200,1,1,2400,192000,1497600,1497600,192000,2400,

%U 1,1,3840,614400,9216000,23961600,9216000,614400,3840,1,1,6480,1658880

%N Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

%C We assume that A059383(0)=1 since it would be the empty product.

%C These are the generalized binomial coefficients associated with the Jordan totient function J_4 given in A059377.

%C Another name might be the 4-totienomial coefficients.

%H Tom Edgar, <a href="http://www.emis.de/journals/INTEGERS/papers/o62/o62.Abstract.html">Totienomial Coefficients</a>, INTEGERS, 14 (2014), #A62.

%H 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 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 T(n,k) = A059383(n)/(A059383(k)* A059383(n-k)).

%F T(n,k) = prod_{i=1..n} A059377(i)/(prod_{i=1..k} A059377(i)*prod_{i=1..n-k} A059377(i)).

%F T(n,k) = A059377(n)/n*(k/A059377(k)*T(n-1,k-1)+(n-k)/A059377(n-k)*T(n-1,k)).

%e The first five terms in the fourth Jordan totient function are 1,15,80,240,624 and so T(4,2) = 240*80*15*1/((15*1)*(15*1))=1280 and T(5,3) = 624*240*80*15*1/((80*15*1)*(15*1))=9984.

%e The triangle begins

%e 1

%e 1 1

%e 1 15 1

%e 1 80 80 1

%e 1 240 1280 240 1

%e 1 624 9984 9984 624 1

%o (Sage)

%o q=100 #change q for more rows

%o P=[0]+[i^4*prod([1-1/p^4 for p in prime_divisors(i)]) for i in [1..q]]

%o [[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 Cf. A059377, A059383, A238453, A238688, A238743.

%K nonn,tabl

%O 0,5

%A _Tom Edgar_, Mar 04 2014

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Last modified October 18 03:25 EDT 2019. Contains 328135 sequences. (Running on oeis4.)