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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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,12
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.
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) = 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)).
EXAMPLE
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.
PROG
(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.
CROSSREFS
Sequence in context: A346529 A238408 A048858 * A172497 A211226 A306366
KEYWORD
nonn,tabl
AUTHOR
Tom Edgar, Aug 27 2014
STATUS
approved

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Last modified April 22 08:40 EDT 2024. Contains 371893 sequences. (Running on oeis4.)