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

Author

Tom Edgar, Mar 22 2014

Keywords

• nonn
• tabl

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.

Examples

			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
		

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. A059378, A059384, A238453, A238688, A238743, A238754.

Programs

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.

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)).