A239702 Triangle read by rows: T(n,k) = A239682(n)/(A239682(k)* A239682(n-k)).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 4, 4, 4, 1, 1, 2, 8, 4, 8, 2, 1, 1, 6, 12, 24, 24, 12, 6, 1, 1, 1, 6, 6, 24, 6, 6, 1, 1, 1, 2, 2, 6, 12, 12, 6, 2, 2, 1, 1, 4, 8, 4, 24, 12, 24, 4, 8, 4, 1, 1, 10, 40, 40, 40, 60, 60, 40, 40, 40, 10, 1, 1, 2

Offset: 0

Author

Tom Edgar, Mar 24 2014

Keywords

• nonn
• tabl

Comments

A239682(0) = 1 since it is the empty product.

These are the generalized binomial coefficients associated with the sequence A173557.

Examples

			The first six terms A173557 are 1,1,2,1,4,2 and so T(4,2) = 1*2*1*1/((1*1)*(1*1))=2 and T(6,3) = 2*4*1*2*1*1/((2*1*1)*(2*1*1))=4.
The triangle begins
1
1 1
1 1  1
1 2  2  1
1 1  2  1  1
1 4  4  4  4  1
1 2  8  4  8  2 1
1 6 12 24 24 12 6 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. A239682, A173557, A048804, A238453.

Programs

Sage

q=100 #change q for more rows
P=[0]+[prod([(x-1) for x in prime_divisors(n)]) for n 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) = A239682(n)/(A239682(k)* A239682(n-k)).

T(n,k) = prod_{i=1..n} A173557(i)/(prod_{i=1..k} A173557(i)*prod_{i=1..n-k} A173557(i)).

T(n,k) = A173557(n)/n*(k/A173557(k)*T(n-1,k-1)+(n-k)/A173557(n-k)*T(n-1,k)).