A129439 - OEIS

%I #14 Sep 12 2024 17:48:55

%S 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,

%T 126,210,210,126,7,1,1,64,224,2688,1680,2688,224,64,1,1,27,864,2016,

%U 9072,9072,2016,864,27,1,1,100,1350,28800,25200,181440,25200,28800,1350,100,1

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

%C 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.

%H G. C. Greubel, <a href="/A129439/b129439.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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)!)).

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

%F T(n, n-k) = T(n, k). - _G. C. Greubel_, Feb 06 2024

%e Triangle starts

%e   1;

%e   1, 1;

%e   1, 2,  1;

%e   1, 3,  3,  1;

%e   1, 8, 12,  8, 1;

%e   1, 5, 20, 20, 5, 1;

%t A092143[n_]:= Product[Floor[n/j]!, {j,n}];

%t A129439[n_, k_]:= A092143[n]/(A092143[n-k]*A092143[k]);

%t Table[A129439[n,k], {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 06 2024 *)

%o (Magma)

%o A092143:= func< n |n eq 0 select 1 else (&*[Factorial(Floor(n/j)): j in [1..n]]) >;

%o A129439:= func< n,k | A092143(n)/(A092143(k)*A092143(n-k)) >;

%o [A129439(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Feb 06 2024

%o (SageMath)

%o def A092143(n): return product(factorial(n//j) for j in range(1,n+1))

%o def A129439(n,k): return A092143(n)//(A092143(n-k)*A092143(k))

%o flatten([[A129439(n,k) for k in range(n+1)] for n in range(14)]) # _G. C. Greubel_, Feb 06 2024

%Y Cf. A007955 (second column), A092143.

%K easy,nonn,tabl

%O 0,5

%A _Peter Bala_, Apr 15 2007