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

0,5

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.

Links

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

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A007955 (second column), A092143.

Sequence in context: A215292 A124975 A171246 * A176469 A141542 A364812

Adjacent sequences: A129436 A129437 A129438 * A129440 A129441 A129442

Keyword

easy,nonn,tabl

Author

Peter Bala, Apr 15 2007

Status

approved