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

0,5

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.

Links

Table of n, a(n) for n=0..52.

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) = 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)).

Example

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

Prog

(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.

Crossrefs

Cf. A059382, A059376, A238688, A238453.

Sequence in context: A168290 A218695 A179837 * A168517 A142465 A248829

Adjacent sequences: A238740 A238741 A238742 * A238744 A238745 A238746

Keyword

nonn,tabl

Author

Tom Edgar, Mar 04 2014

Status

approved