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A141605 Triangle read by rows: T(n, k) = A006722(n)/(A006722(k)*A006722(n-k)).

%I A141605 #18 Sep 22 2024 07:24:19
%S A141605 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,1,1,2,5,5,5,5,
%T A141605 2,1,1,2,3,9,9,9,3,2,1,1,3,5,8,23,23,8,5,3,1,1,3,8,15,25,75,25,15,8,3,
%U A141605 1,1,6,18,47,84,140,140,84,47,18,6,1
%N A141605 Triangle read by rows: T(n, k) = A006722(n)/(A006722(k)*A006722(n-k)).
%H A141605 G. C. Greubel, <a href="/A141605/b141605.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A141605 T(n, k) = A006722(n)/(A006722(n-k)*A006722(m)).
%e A141605 Triangle begins as:
%e A141605   1;
%e A141605   1,  1;
%e A141605   1,  1,  1;
%e A141605   1,  1,  1,  1;
%e A141605   1,  1,  1,  1,  1;
%e A141605   1,  1,  1,  1,  1,  1;
%e A141605   1,  3,  3,  3,  3,  3,  1;
%e A141605   1,  2,  5,  5,  5,  5,  2,  1;
%e A141605   1,  2,  3,  9,  9,  9,  3,  2,  1;
%e A141605   1,  3,  5,  8, 23, 23,  8,  5,  3,  1;
%e A141605   1,  3,  8, 15, 25, 75, 25, 15,  8,  3,  1;
%e A141605   ...
%t A141605 f[n_]:= f[n]= If[n<6, 1, (f[n-1]*f[n-5] +f[n-2]*f[n-4] +f[n-3]^2)/f[n-6]] (*A006722*)
%t A141605 A141605[n_, k_]:= Round[f[n]/(f[k]*f[n-k])];
%t A141605 Table[A141605[n, k], {n,0,12}, {k,0,n}]//Flatten
%o A141605 (Magma)
%o A141605 A006722:= [n le 6 select 1 else (Self(n-1)*Self(n-5) + Self(n-2)*Self(n-4) + Self(n-3)^2)/Self(n-6): n in [1..30]];
%o A141605 A141605:= func< n, k | Round(A006722[n+1]/(A006722[k+1]*A006722[n-k+1])) >;
%o A141605 [A141605(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 21 2024
%o A141605 (SageMath)
%o A141605 def f(n): # f = A006722
%o A141605     if n<6: return 1
%o A141605     else: return (f(n-1)*f(n-5) +f(n-2)*f(n-4) +f(n-3)^2)/f(n-6)
%o A141605 def A141605(n, k): return round(f(n)/(f(k)*f(n-k)))
%o A141605 flatten([[A141605(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 21 2024
%Y A141605 Cf. A006722, A141604.
%K A141605 nonn,tabl
%O A141605 0,23
%A A141605 _Roger L. Bagula_ and _Gary W. Adamson_, Aug 21 2008
%E A141605 Edited and new name by _G. C. Greubel_, Sep 21 2024