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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

Table of n, a(n) for n=0..84.

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

Cf. A003557, A085056, A048804.

Sequence in context: A126067 A238408 A048858 * A172497 A211226 A135265

Adjacent sequences:  A246462 A246463 A246464 * A246466 A246467 A246468

KEYWORD

nonn,tabl

AUTHOR

Tom Edgar, Aug 27 2014

STATUS

approved

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Last modified October 18 18:58 EDT 2018. Contains 316323 sequences. (Running on oeis4.)