A238453 Triangle read by rows: T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).
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
Examples
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
Links
- Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
- 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.
Programs
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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 -
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 *) -
PARI
T(n,k)={prod(i=1, k, eulerphi(n+1-i)/eulerphi(i))} \\ Andrew Howroyd, Nov 13 2018 -
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.
Formula
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
Comments