A129455 An analog of Pascal's triangle based on A129454. T(n, k) = A129454(n+1)/(A129454(n-k+1)*A129454(k+1)).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 256, 384, 256, 1, 1, 5, 640, 640, 5, 1, 1, 1146617856, 2866544640, 244611809280, 2866544640, 1146617856, 1, 1, 7, 4013162496, 6688604160, 6688604160, 4013162496, 7, 1, 1, 35184372088832, 123145302310912, 47066867504069920948224, 919274755938865643520, 47066867504069920948224, 123145302310912, 35184372088832, 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 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

A129454[n_]:= Product[GCD[j, k, m], {j, n-1}, {k, n-1}, {m, n-1}];
A129455[n_, k_]:= A129454[n+1]/(A129454[k+1]*A129454[n-k+1]);
Table[A129455[n, k], {n, 0, 9}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 07 2024 *)

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:= func< n, k | A129454(n+1)/(A129454(n-k+1)*A129454(k+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))
def A129455(n, k): return A129454(n+1)/(A129454(n-k+1)*A129454(k+1))
flatten([[A129455(n, k) for k in range(n+1)] for n in range(10)]) # G. C. Greubel, Feb 07 2024

Crossrefs

Cf. A092287, A129453, A129454.

Sequence in context: A141542 A364812 A129453 * A329322 A067924 A344912

Adjacent sequences: A129452 A129453 A129454 * A129456 A129457 A129458

Keyword

nonn,tabl

Author

Peter Bala, Apr 16 2007

Status

approved