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URL: https://oeis.org/A239702

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A239702
Triangle read by rows: T(n,k) = A239682(n)/(A239682(k)* A239682(n-k)).
0
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
0,8
COMMENTS
A239682(0) = 1 since it is the empty product.
These are the generalized binomial coefficients associated with the sequence A173557.
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.
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)).
EXAMPLE
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
PROG
(SageMath)
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.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tom Edgar, Mar 24 2014
STATUS
approved