A238743 Triangle read by rows: T(n,k) = A059382(n)/(A059382(k)*A059382(n-k)).

1, 1, 1, 1, 7, 1, 1, 26, 26, 1, 1, 56, 208, 56, 1, 1, 124, 992, 992, 124, 1, 1, 182, 3224, 6944, 3224, 182, 1, 1, 342, 8892, 42408, 42408, 8892, 342, 1, 1, 448, 21888, 153216, 339264, 153216, 21888, 448, 1, 1, 702, 44928, 590976, 1920672, 1920672, 590976, 44928

Offset: 0

Author

Tom Edgar, Mar 04 2014

Keywords

• nonn
• tabl

Comments

We assume that A059382(0)=1 since it would be the empty product.

These are the generalized binomial coefficients associated with the Jordan totient function J_3 given in A059376.

Another name might be the 3-totienomial coefficients.

Examples

			The first five terms in the third Jordan totient function are 1,7,26,56,124 and so T(4,2) = 56*26*7*1/((7*1)*(7*1))=208 and T(5,3) = 124*56*26*7*1/((26*7*1)*(7*1))=992.
The triangle begins
1
1 1
1 7   1
1 26  26   1
1 56  208  56   1
1 124 992  992  124  1
1 182 3224 6944 3224 182 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. A059382, A059376, A238688, A238453.

Programs

Sage

q=100 #change q for more rows
P=[0]+[i^3*prod([1-1/p^3 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) = A059382(n)/(A059382(k)* A059382(n-k)).

T(n,k) = prod_{i=1..n} A059376(i)/(prod_{i=1..k} A059376(i)*prod_{i=1..n-k} A059376(i)).

T(n,k) = A059376(n)/n*(k/A059376(k)*T(n-1,k-1)+(n-k)/A059376(n-k)*T(n-1,k)).