A129453 An analog of Pascal's triangle based on A092287. T(n,k) = A092287(n)/(A092287(n-k)*A092287(k)), 0 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 16, 24, 16, 1, 1, 5, 40, 40, 5, 1, 1, 864, 2160, 11520, 2160, 864, 1, 1, 7, 3024, 5040, 5040, 3024, 7, 1, 1, 2048, 7168, 2064384, 645120, 2064384, 7168, 2048, 1, 1, 729, 746496, 1741824, 94058496, 94058496, 1741824, 746496, 729, 1

Offset

0,5

Comments

It appears that the T(n,k) are always integers. This would follow from the conjectured prime factorization given in A092287. Calculation suggests that the binomial coefficients C(n,k) divide T(n,k) and indeed that T(n,k)/C(n,k) are perfect squares.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = (Product_{i=1..n} Product_{j=1..n} gcd(i,j)) / ( (Product_{i=1..n-k} Product_{j=1..n-k} gcd(i,j)) * ( Product_{i=1..k} Product_{j=1..k} gcd(i,j)) ), note that empty products equal to 1.

T(n, n-k) = T(n, k). - G. C. Greubel, Feb 07 2024

Example

Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1, 16, 24, 16,  1;
  1,  5, 40, 40,  5,  1;

Mathematica

A092287[n_]:= Product[GCD[j, k], {j, n}, {k, n}];
A129453[n_, k_]:= A092287[n]/(A092287[k]*A092287[n-k]);
Table[A129453[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 07 2024 *)

Prog

(Magma)
A092287:= func< n | n eq 0 select 1 else (&*[(&*[GCD(j, k): k in [1..n]]): j in [1..n]]) >;
A129453:= func< n, k | A092287(n)/(A092287(n-k)*A092287(k)) >;
[A129453(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 07 2024
(SageMath)
def A092287(n): return product(product( gcd(j, k) for k in range(1, n+1)) for j in range(1, n+1))
def A129453(n, k): return A092287(n)/(A092287(n-k)*A092287(k))
flatten([[A129453(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 07 2024

Crossrefs

Cf. A007318, A092287, A129454, A129455.

Sequence in context: A176469 A141542 A364812 * A129455 A329322 A067924

Adjacent sequences: A129450 A129451 A129452 * A129454 A129455 A129456

Keyword

nonn,tabl

Author

Peter Bala, Apr 16 2007

Status

approved