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

Author

Tom Edgar, Feb 27 2014

Keywords

• nonn
• tabl

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.

Examples

			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
		

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.

Crossrefs

Cf. A001615, A175836, A238453.

Programs

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

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 *)

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.

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).