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URL: https://oeis.org/A238453

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A238453
Triangle read by rows: T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).
12
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 8, 8, 4, 1, 1, 2, 8, 8, 8, 2, 1, 1, 6, 12, 24, 24, 12, 6, 1, 1, 4, 24, 24, 48, 24, 24, 4, 1, 1, 6, 24, 72, 72, 72, 72, 24, 6, 1, 1, 4, 24, 48, 144, 72, 144, 48, 24, 4, 1, 1, 10, 40, 120, 240, 360, 360, 240, 120
OFFSET
0,8
COMMENTS
We assume that A001088(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with Euler's totient function A000010.
Another name might be the totienomial coefficients.
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) = A001088(n)/(A001088(k)*A001088(n-k)).
T(n,k) = prod_{i=1..n} A000010(i)/(prod_{i=1..k} A000010(i)*prod_{i=1..n-k} A000010(i)).
T(n,k) = A000010(n)/n*(k/A000010(k)*T(n-1,k-1)+(n-k)/A000010(n-k)*T(n-1,k)).
T(n+1, 2) = A083542(n). - Michael Somos, Aug 26 2014
T(n,k) = Product_{i=1..k} (phi(n+1-i)/phi(i)), where phi is Euler's totient function (A000010). - Werner Schulte, Nov 14 2018
EXAMPLE
The first five terms in Euler's totient function are 1,1,2,2,4 and so T(4,2) = 2*2*1*1/((1*1)*(1*1))=4 and T(5,3) = 4*2*2*1*1/((2*1*1)*(1*1))=8.
The triangle begins
1
1 1
1 1 1
1 2 2 1
1 2 4 2 1
1 4 8 8 4 1
1 2 8 8 8 2 1
MATHEMATICA
f[n_] := Product[EulerPhi@ k, {k, n}]; Table[f[n]/(f[k] f[n - k]), {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 19 2016 *)
PROG
(SageMath)
q=100 #change q for more rows
P=[euler_phi(i) for i in [0..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.
(Haskell)
a238453 n k = a238453_tabl !! n !! k
a238453_row n = a238453_tabl !! n
a238453_tabl = [1] : f [1] a000010_list where
f xs (z:zs) = (map (div y) $ zipWith (*) ys $ reverse ys) : f ys zs
where ys = y : xs; y = head xs * z
-- Reinhard Zumkeller, Feb 27 2014
(PARI) T(n, k)={prod(i=1, k, eulerphi(n+1-i)/eulerphi(i))} \\ Andrew Howroyd, Nov 13 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Tom Edgar, Feb 26 2014
STATUS
approved