A255914 - OEIS

%I #12 Jan 25 2016 14:23:38

%S 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 1,42,252,840,840,252,42,1,1,32,672,1344,3360,1344,672,32,1,1,54,864,

%U 6048,9072,9072,6048,864,54,1,1,40,1080,5760,30240,18144,30240

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

%C These are the generalized binomial coefficients associated with the sequence A002618.

%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.

%F T(n,k) = Product_{i=1..n} A002618(i)/(Product_{i=1..k} A002618(i)*Product_{i=1..n-k} A002618(i)).

%F T(n,k) = A002618(n)/n*(k/A002618(k)*T(n-1,k-1)+(n-k)/A002618(n-k)*T(n-1,k)).

%e 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 The triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 6, 6, 1;

%e 1, 8, 24, 8, 1;

%e 1, 20, 80, 80, 20, 1;

%e 1, 12, 120, 160, 120, 12, 1;

%e 1, 42, 252, 840, 840, 252, 42, 1

%o (Sage)

%o q=100 #change q for more rows

%o P=[i*euler_phi(i) for i in [0..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. A002618, A238453, A007318.

%K nonn,tabl

%O 0,5

%A _Tom Edgar_, Mar 10 2015