A246465 Triangle read by rows: T(n,k) = A085056(n)/(A085056(k) * A085056(n-k)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 2, 4, 4, 4, 1, 1, 3, 12, 12, 6, 6, 12, 12, 3, 1, 1, 1, 3, 12, 6, 6, 6, 12, 3, 1, 1, 1, 1, 1, 3, 6, 6, 6, 6, 3, 1, 1, 1, 1, 2, 2, 2, 3, 12, 12

Offset: 0

Author

Tom Edgar, Aug 27 2014

Keywords

• nonn
• tabl

Comments

We assume that A085056(0)=1 since it would be the empty product.

These are the generalized binomial coefficients associated with the sequence A003557.

Examples

			The first five terms in A003557 are: 1, 1, 1, 2, 1 and so T(4,2) = 2*1*1*1/((1*1)*(1*1))=2 and T(5,4) = 1*2*1*1*1/((2*1*1*1)*(1))=1.
The triangle begins:
1,
1, 1,
1, 1, 1,
1, 1, 1, 1,
1, 2, 2, 2, 1,
1, 1, 2, 2, 1, 1,
1, 1, 1, 2, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1,
1, 4, 4, 4, 2, 4, 4, 4, 1,
1, 3, 12, 12, 6, 6, 12, 12, 3, 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. A003557, A085056, A048804.

Programs

Sage

q=100 #change q for more rows
P=[0]+[n/prod([x 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) = A085056(n)/(A085056(k) * A085056(n-k)).

T(n,k) = prod_{i=1..n} A003557(i)/(prod_{i=1..k} A003557(i)*prod_{i=1..n-k} A003557(i)).

T(n,k) = A003557(n)/n*(k/A003557(k)*T(n-1,k-1)+(n-k)/A003557(n-k)*T(n-1,k)).