OFFSET
0,5
COMMENTS
It appears that the T(n,k) are always integers. This would follow from the conjectured prime factorization given in A129454. Calculation suggests that the binomial coefficients C(n,k) divide T(n,k) and that T(n,k)/C(n,k) are perfect sixth powers.
LINKS
G. C. Greubel, Rows n = 0..25 of the triangle, flattened
FORMULA
T(n, k) = Product_{h=1..n} Product_{i=1..n} Product_{j=1..n} gcd(h,i,j)/( (Product_{h=1..n-k} Product_{i=1..n-k} Product_{j=1..n-k} gcd(h,i,j))*(Product_{h=1..k} Product_{i=1..k} Product_{j=1..k} gcd(h,i,j)) ).
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, 256, 384, 256, 1;
1, 5, 640, 640, 5, 1;
MATHEMATICA
PROG
(Magma)
A129454:= func< n | n le 1 select 1 else (&*[(&*[(&*[GCD(GCD(j, k), m): k in [1..n-1]]): j in [1..n-1]]): m in [1..n-1]]) >;
[A129455(n, k): k in [0..n], n in [0..9]]; // G. C. Greubel, Feb 07 2024
(SageMath)
def A129454(n): return product(product(product(gcd(gcd(j, k), m) for k in range(1, n)) for j in range(1, n)) for m in range(1, n))
flatten([[A129455(n, k) for k in range(n+1)] for n in range(10)]) # G. C. Greubel, Feb 07 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Bala, Apr 16 2007
STATUS
approved