A239702 - OEIS

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

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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

Table of n, a(n) for n=0..79.

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 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

(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.

CROSSREFS

Cf. A239682, A173557, A048804, A238453.

Sequence in context: A030616 A066422 A274888 * A202084 A337763 A109072

Adjacent sequences: A239699 A239700 A239701 * A239703 A239704 A239705

KEYWORD

nonn,tabl

AUTHOR

Tom Edgar, Mar 24 2014

STATUS

approved