login
Triangle read by rows: T(n,k) = A059382(n)/(A059382(k)*A059382(n-k)).
3

%I #19 Jan 25 2016 14:22:05

%S 1,1,1,1,7,1,1,26,26,1,1,56,208,56,1,1,124,992,992,124,1,1,182,3224,

%T 6944,3224,182,1,1,342,8892,42408,42408,8892,342,1,1,448,21888,153216,

%U 339264,153216,21888,448,1,1,702,44928,590976,1920672,1920672,590976,44928

%N Triangle read by rows: T(n,k) = A059382(n)/(A059382(k)*A059382(n-k)).

%C We assume that A059382(0)=1 since it would be the empty product.

%C These are the generalized binomial coefficients associated with the Jordan totient function J_3 given in A059376.

%C Another name might be the 3-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) = A059382(n)/(A059382(k)* A059382(n-k)).

%F T(n,k) = prod_{i=1..n} A059376(i)/(prod_{i=1..k} A059376(i)*prod_{i=1..n-k} A059376(i)).

%F T(n,k) = A059376(n)/n*(k/A059376(k)*T(n-1,k-1)+(n-k)/A059376(n-k)*T(n-1,k)).

%e The first five terms in the third Jordan totient function are 1,7,26,56,124 and so T(4,2) = 56*26*7*1/((7*1)*(7*1))=208 and T(5,3) = 124*56*26*7*1/((26*7*1)*(7*1))=992.

%e The triangle begins

%e 1

%e 1 1

%e 1 7 1

%e 1 26 26 1

%e 1 56 208 56 1

%e 1 124 992 992 124 1

%e 1 182 3224 6944 3224 182 1

%o (Sage)

%o q=100 #change q for more rows

%o P=[0]+[i^3*prod([1-1/p^3 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. A059382, A059376, A238688, A238453.

%K nonn,tabl

%O 0,5

%A _Tom Edgar_, Mar 04 2014