A129453 - OEIS

%I #9 Feb 07 2024 02:56:48

%S 1,1,1,1,2,1,1,3,3,1,1,16,24,16,1,1,5,40,40,5,1,1,864,2160,11520,2160,

%T 864,1,1,7,3024,5040,5040,3024,7,1,1,2048,7168,2064384,645120,2064384,

%U 7168,2048,1,1,729,746496,1741824,94058496,94058496,1741824,746496,729,1

%N An analog of Pascal's triangle based on A092287. T(n,k) = A092287(n)/(A092287(n-k)*A092287(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 A092287. Calculation suggests that the binomial coefficients C(n,k) divide T(n,k) and indeed that T(n,k)/C(n,k) are perfect squares.

%H G. C. Greubel, <a href="/A129453/b129453.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = (Product_{i=1..n} Product_{j=1..n} gcd(i,j)) / ( (Product_{i=1..n-k} Product_{j=1..n-k} gcd(i,j)) * ( Product_{i=1..k} Product_{j=1..k} gcd(i,j)) ), note that empty products equal to 1.

%F T(n, n-k) = T(n, k). - _G. C. Greubel_, Feb 07 2024

%e Triangle starts:

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  3,  3,  1;

%e   1, 16, 24, 16,  1;

%e   1,  5, 40, 40,  5,  1;

%t A092287[n_]:= Product[GCD[j,k], {j,n}, {k,n}];

%t A129453[n_, k_]:= A092287[n]/(A092287[k]*A092287[n-k]);

%t Table[A129453[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 07 2024 *)

%o (Magma)

%o A092287:= func< n | n eq 0 select 1 else (&*[(&*[GCD(j,k): k in [1..n]]): j in [1..n]]) >;

%o A129453:= func< n,k | A092287(n)/(A092287(n-k)*A092287(k)) >;

%o [A129453(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 07 2024

%o (SageMath)

%o def A092287(n): return product(product( gcd(j,k) for k in range(1,n+1)) for j in range(1,n+1))

%o def A129453(n,k): return A092287(n)/(A092287(n-k)*A092287(k))

%o flatten([[A129453(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 07 2024

%Y Cf. A007318, A092287, A129454, A129455.

%K nonn,tabl

%O 0,5

%A _Peter Bala_, Apr 16 2007