A238754 Triangle read by rows: T(n,k) = A059383(n)/(A059383(k)*A059383(n-k)).

1, 1, 1, 1, 15, 1, 1, 80, 80, 1, 1, 240, 1280, 240, 1, 1, 624, 9984, 9984, 624, 1, 1, 1200, 49920, 149760, 49920, 1200, 1, 1, 2400, 192000, 1497600, 1497600, 192000, 2400, 1, 1, 3840, 614400, 9216000, 23961600, 9216000, 614400, 3840, 1, 1, 6480, 1658880

Offset: 0

Author

Tom Edgar, Mar 04 2014

Keywords

• nonn
• tabl

Comments

We assume that A059383(0)=1 since it would be the empty product.

These are the generalized binomial coefficients associated with the Jordan totient function J_4 given in A059377.

Another name might be the 4-totienomial coefficients.

Examples

			The first five terms in the fourth Jordan totient function are 1,15,80,240,624 and so T(4,2) = 240*80*15*1/((15*1)*(15*1))=1280 and T(5,3) = 624*240*80*15*1/((80*15*1)*(15*1))=9984.
The triangle begins
1
1 1
1 15  1
1 80  80   1
1 240 1280 240  1
1 624 9984 9984 624 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. A059377, A059383, A238453, A238688, A238743.

Programs

Sage

q=100 #change q for more rows
P=[0]+[i^4*prod([1-1/p^4 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) = A059383(n)/(A059383(k)* A059383(n-k)).

T(n,k) = prod_{i=1..n} A059377(i)/(prod_{i=1..k} A059377(i)*prod_{i=1..n-k} A059377(i)).

T(n,k) = A059377(n)/n*(k/A059377(k)*T(n-1,k-1)+(n-k)/A059377(n-k)*T(n-1,k)).