A238688 Triangle read by rows: T(n,k) = A059381(n)/(A059381(k)*A059381(n-k)).

1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 12, 32, 12, 1, 1, 24, 96, 96, 24, 1, 1, 24, 192, 288, 192, 24, 1, 1, 48, 384, 1152, 1152, 384, 48, 1, 1, 48, 768, 2304, 4608, 2304, 768, 48, 1, 1, 72, 1152, 6912, 13824, 13824, 6912, 1152, 72, 1, 1, 72, 1728, 10368, 41472, 41472

Offset

0,5

Comments

We assume that A059381(0)=1 since it would be the empty product.

These are the generalized binomial coefficients associated with the Jordan totient function J_2 given in A007434.

Another name might be the 2-totienomial coefficients.

Links

Table of n, a(n) for n=0..60.

Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.

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.

Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Formula

T(n,k) = A059381(n)/(A059381(k)* A059381(n-k)).

T(n,k) = prod_{i=1..n} A007434(i)/(prod_{i=1..k} A007434(i)*prod_{i=1..n-k} A007434(i)).

T(n,k) = A007434(n)/n*(k/A007434(k)*T(n-1,k-1)+(n-k)/A007434(n-k)*T(n-1,k)).

Example

The first five terms in the second Jordan totient function are 1,3,8,12,24 and so T(4,2) = 12*8*3*1/((3*1)*(3*1))=32 and T(5,3) = 24*12*8*3*1/((8*3*1)*(3*1))=96.
The triangle begins
1
1   1
1   3   1
1   8   8   1
1   12  32  12   1
1   24  96  96   24   1
1   24  192 288  192  24 1

Prog

(Sage)
q=100 #change q for more rows
P=[0]+[i^2*prod([1-1/p^2 for p in prime_divisors(i)]) for i in [1..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. A007434, A059381, A238453.

Sequence in context: A124469 A094816 A097712 * A174117 A157210 A359573

Adjacent sequences: A238685 A238686 A238687 * A238689 A238690 A238691

Keyword

nonn,tabl

Author

Tom Edgar, Mar 02 2014

Status

approved