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

0,8

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

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.

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

(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
-- 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

Cf. A000010, A001088, A083542.

Sequence in context: A122085 A209612 A209805 * A066287 A059260 A239473

Adjacent sequences: A238450 A238451 A238452 * A238454 A238455 A238456

Keyword

nonn,tabl

Author

Tom Edgar, Feb 26 2014

Status

approved