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

0,5

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.

Links

Table of n, a(n) for n=0..47.

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) = 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)).

Example

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

Prog

(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.

Crossrefs

Cf. A059377, A059383, A238453, A238688, A238743.

Sequence in context: A174389 A176286 A111805 * A176226 A155493 A156939

Adjacent sequences: A238751 A238752 A238753 * A238755 A238756 A238757

Keyword

nonn,tabl

Author

Tom Edgar, Mar 04 2014

Status

approved