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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
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OFFSET
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0,8
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Sage)
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
(PARI) T(n, k)={prod(i=1, k, eulerphi(n+1-i)/eulerphi(i))} \\ Andrew Howroyd, Nov 13 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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