A141605 Triangle read by rows: T(n, k) = A006722(n)/(A006722(k)*A006722(n-k)).
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, 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, 1, 1, 6, 18, 47, 84, 140, 140, 84, 47, 18, 6, 1
Offset: 0
Examples
Triangle begins as: 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, 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, 1; ...
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Magma
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]]; A141605:= func< n, k | Round(A006722[n+1]/(A006722[k+1]*A006722[n-k+1])) >; [A141605(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 21 2024
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Mathematica
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*) A141605[n_, k_]:= Round[f[n]/(f[k]*f[n-k])]; Table[A141605[n, k], {n,0,12}, {k,0,n}]//Flatten
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SageMath
def f(n): # f = A006722 if n<6: return 1 else: return (f(n-1)*f(n-5) +f(n-2)*f(n-4) +f(n-3)^2)/f(n-6) def A141605(n, k): return round(f(n)/(f(k)*f(n-k))) flatten([[A141605(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 21 2024
Extensions
Edited and new name by G. C. Greubel, Sep 21 2024