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

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, 1, 42, 252, 840, 840, 252, 42, 1, 1, 32, 672, 1344, 3360, 1344, 672, 32, 1, 1, 54, 864, 6048, 9072, 9072, 6048, 864, 54, 1, 1, 40, 1080, 5760, 30240, 18144, 30240

Offset: 0

Author

Tom Edgar, Mar 10 2015

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
The triangle begins:
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;
1, 42, 252, 840, 840, 252, 42, 1
		

Links

• Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Crossrefs

Cf. A002618, A238453, A007318.

Programs

Sage

q=100 #change q for more rows
P=[i*euler_phi(i) for i in [0..q]]
[[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.

Formula

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

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