A239633 Triangle read by rows: T(n,k) = A059384(n)/(A059384(k)*A059384(n-k)).

1, 1, 1, 1, 31, 1, 1, 242, 242, 1, 1, 992, 7744, 992, 1, 1, 3124, 99968, 99968, 3124, 1, 1, 7502, 756008, 3099008, 756008, 7502, 1, 1, 16806, 4067052, 52501944, 52501944, 4067052, 16806, 1, 1, 31744, 17209344, 533489664, 1680062208, 533489664, 17209344, 31744

Offset

0,5

Comments

We assume that A059384(0)=1 since it would be the empty product.

These are the generalized binomial coefficients associated with the Jordan totient function J_5 given in A059378.

Another name might be the 5-totienomial coefficients.

Links

Table of n, a(n) for n=0..43.

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) = A059384(n)/(A059384(k)* A059384(n-k)).

T(n,k) = prod_{i=1..n} A059378(i)/(prod_{i=1..k} A059378(i)*prod_{i=1..n-k} A059378(i)).

T(n,k) = A059378(n)/n*(k/A059378(k)*T(n-1,k-1)+(n-k)/A059378(n-k)*T(n-1,k)).

Example

The first five terms in the fifth Jordan totient function are 1,31,242,992,3124 and so T(4,2) = 992*242*31*1/((31*1)*(31*1))=7744 and T(5,3) = 3124*992*242*31*1/((242*31*1)*(31*1))=99968.
The triangle begins
1
1 1
1 31   1
1 242  242   1
1 992  7744  992   1
1 3124 99968 99968 3124 1

Prog

(Sage)
q=100 #change q for more rows
P=[0]+[i^5*prod([1-1/p^5 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. A059378, A059384, A238453, A238688, A238743, A238754.

Sequence in context: A040962 A040961 A300656 * A174692 A172302 A103474

Adjacent sequences: A239630 A239631 A239632 * A239634 A239635 A239636

Keyword

nonn,tabl

Author

Tom Edgar, Mar 22 2014

Status

approved