A129439 An analog of Pascal's triangle: T(n,k) = A092143(n)/(A092143(n-k)*A092143(k)), 0 <= k <= n.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 8, 12, 8, 1, 1, 5, 20, 20, 5, 1, 1, 36, 90, 240, 90, 36, 1, 1, 7, 126, 210, 210, 126, 7, 1, 1, 64, 224, 2688, 1680, 2688, 224, 64, 1, 1, 27, 864, 2016, 9072, 9072, 2016, 864, 27, 1, 1, 100, 1350, 28800, 25200, 181440, 25200, 28800, 1350, 100, 1

Offset: 0

Author

Peter Bala, Apr 15 2007

Keywords

• easy
• nonn
• tabl

Comments

It appears that the T(n,k) are always integers. This would follow from the conjectured prime factorization given in the comments section of A092143.

Examples

			Triangle starts
  1;
  1, 1;
  1, 2,  1;
  1, 3,  3,  1;
  1, 8, 12,  8, 1;
  1, 5, 20, 20, 5, 1;
		

Links

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

Crossrefs

Cf. A007955 (second column), A092143.

Programs

Magma

A092143:= func< n |n eq 0 select 1 else (&*[Factorial(Floor(n/j)): j in [1..n]]) >;
A129439:= func< n,k | A092143(n)/(A092143(k)*A092143(n-k)) >;
[A129439(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Feb 06 2024

Mathematica

A092143[n_]:= Product[Floor[n/j]!, {j,n}];
A129439[n_, k_]:= A092143[n]/(A092143[n-k]*A092143[k]);
Table[A129439[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 06 2024 *)

SageMath

def A092143(n): return product(factorial(n//j) for j in range(1,n+1))
def A129439(n,k): return A092143(n)//(A092143(n-k)*A092143(k))
flatten([[A129439(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Feb 06 2024

Formula

T(n,k) = Product_{j=1..n} floor(n/j)!/((Product_{j=1..n-k} floor((n-k)/j)!)*(Product_{j=1..k} floor(k/j)!)).

T(n, 1) = A007955(n).

T(n, n-k) = T(n, k). - G. C. Greubel, Feb 06 2024