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A129439
An analog of Pascal's triangle: T(n,k) = A092143(n)/(A092143(n-k)*A092143(k)), 0 <= k <= n.
3
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.
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
KEYWORD
easy,nonn,tabl
AUTHOR
Peter Bala, Apr 15 2007
STATUS
approved