%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
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