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

Author

Tom Edgar, Mar 02 2014

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• 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.

Crossrefs

Cf. A007434, A059381, A238453.

Programs

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.

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)).