A129455 - OEIS

%I #9 Feb 08 2024 01:45:20

%S 1,1,1,1,2,1,1,3,3,1,1,256,384,256,1,1,5,640,640,5,1,1,1146617856,

%T 2866544640,244611809280,2866544640,1146617856,1,1,7,4013162496,

%U 6688604160,6688604160,4013162496,7,1,1,35184372088832,123145302310912,47066867504069920948224,919274755938865643520,47066867504069920948224,123145302310912,35184372088832,1

%N An analog of Pascal's triangle based on A129454. T(n, k) = A129454(n+1)/(A129454(n-k+1)*A129454(k+1)).

%C It appears that the T(n,k) are always integers. This would follow from the conjectured prime factorization given in A129454. Calculation suggests that the binomial coefficients C(n,k) divide T(n,k) and that T(n,k)/C(n,k) are perfect sixth powers.

%H G. C. Greubel, <a href="/A129455/b129455.txt">Rows n = 0..25 of the triangle, flattened</a>

%F T(n, k) = Product_{h=1..n} Product_{i=1..n} Product_{j=1..n} gcd(h,i,j)/( (Product_{h=1..n-k} Product_{i=1..n-k} Product_{j=1..n-k} gcd(h,i,j))*(Product_{h=1..k} Product_{i=1..k} Product_{j=1..k} gcd(h,i,j)) ).

%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, 256, 384, 256,  1;

%e   1,   5, 640, 640,  5,  1;

%t A129454[n_]:= Product[GCD[j,k,m], {j,n-1}, {k,n-1}, {m,n-1}];

%t A129455[n_, k_]:= A129454[n+1]/(A129454[k+1]*A129454[n-k+1]);

%t Table[A129455[n,k], {n,0,9}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 07 2024 *)

%o (Magma)

%o A129454:= func< n | n le 1 select 1 else (&*[(&*[(&*[GCD(GCD(j,k),m): k in [1..n-1]]): j in [1..n-1]]): m in [1..n-1]]) >;

%o A129455:= func< n,k | A129454(n+1)/(A129454(n-k+1)*A129454(k+1)) >;

%o [A129455(n,k): k in [0..n], n in [0..9]]; // _G. C. Greubel_, Feb 07 2024

%o (SageMath)

%o def A129454(n): return product(product(product(gcd(gcd(j,k),m) for k in range(1,n)) for j in range(1,n)) for m in range(1,n))

%o def A129455(n,k): return A129454(n+1)/(A129454(n-k+1)*A129454(k+1))

%o flatten([[A129455(n,k) for k in range(n+1)] for n in range(10)]) # _G. C. Greubel_, Feb 07 2024

%Y Cf. A092287, A129453, A129454.

%K nonn,tabl

%O 0,5

%A _Peter Bala_, Apr 16 2007