A238498 Triangle read by rows: T(n,k) = A175836(n)/(A175836(k)* A175836(n-k)).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 8, 6, 1, 1, 6, 12, 12, 6, 1, 1, 12, 24, 36, 24, 12, 1, 1, 8, 32, 48, 48, 32, 8, 1, 1, 12, 32, 96, 96, 96, 32, 12, 1, 1, 12, 48, 96, 192, 192, 96, 48, 12, 1, 1, 18, 72, 216, 288, 576, 288, 216, 72, 18, 1, 1, 12, 72, 216, 432, 576, 576, 432, 216, 72, 12, 1

Offset

0,5

Comments

We assume that A175836(0)=1 since it would be the empty product.

These are the generalized binomial coefficients associated with the Dedekind psi function A001615.

Another name might be the psi-nomial 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) = A175836(n)/(A175836(k)*A175836(n-k)).

T(n,k) = prod_{i=1..n} A001615(i)/(prod_{i=1..k} A001615(i)*prod_{i=1..n-k} A001615(i)).

T(n,k) = A001615(n)/n*(k/A001615(k)*T(n-1,k-1)+(n-k)/A001615(n-k)*T(n-1,k)).

T(n,k) = A238688(n,k)/A238453(n,k).

Example

The first five terms in the Dedekind psi function are 1,3,4,6,6 and so T(4,2) = 6*4*3*1/((3*1)*(3*1))=8 and T(5,3) = 6*6*4*3*1/((4*3*1)*(3*1))=12.
The triangle begins
1
1  1
1  3  1
1  4  4  1
1  6  8  6  1
1  6  12 12 6 1

Maple

A175836 := proc(n) option remember; local p;
`if`(n<2, 1, n*mul(1+1/p, p=factorset(n))*A175836(n-1)) end:
A238498 := (n, k) -> A175836(n)/(A175836(k)*A175836(n-k)):
seq(seq(A238498(n, k), k=0..n), n=0..10); # Peter Luschny, Feb 28 2014

Mathematica

DedekindPsi[n_] := Sum[MoebiusMu[n/d] d^2 , {d, Divisors[n]}]/EulerPhi[n];
(* b = A175836 *) b[n_] := Times @@ DedekindPsi /@ Range[n];
T[n_, k_] := b[n]/(b[k] b[n-k]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jul 02 2019 *)

Prog

(Sage)
q=100 #change q for more rows
P=[0]+[i*prod([(1+1/x) for x in prime_divisors(i)]) for i 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.
(Haskell)
a238498 n k = a238498_tabl !! n !! k
a238498_row n = a238498_tabl !! n
a238498_tabl = [1] : f [1] a001615_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, Mar 01 2014

Crossrefs

Cf. A001615, A175836, A238453.

Sequence in context: A147290 A026670 A131402 * A026626 A136482 A026648

Adjacent sequences: A238495 A238496 A238497 * A238499 A238500 A238501

Keyword

nonn,tabl

Author

Tom Edgar, Feb 27 2014

Status

approved