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

0,5

Comments

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

Links

Table of n, a(n) for n=0..61.

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.

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)).

Example

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

Prog

(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.

Crossrefs

Cf. A002618, A238453, A007318.

Sequence in context: A260238 A283795 A168641 * A143185 A347675 A157635

Adjacent sequences: A255911 A255912 A255913 * A255915 A255916 A255917

Keyword

nonn,tabl

Author

Tom Edgar, Mar 10 2015

Status

approved