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

0,23

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = A006722(n)/(A006722(n-k)*A006722(m)).

Example

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;
  ...

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

Prog

(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
(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

Crossrefs

Cf. A006722, A141604.

Sequence in context: A247655 A097675 A381930 * A356002 A251551 A073139

Adjacent sequences: A141602 A141603 A141604 * A141606 A141607 A141608

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Aug 21 2008

Extensions

Edited and new name by G. C. Greubel, Sep 21 2024

Status

approved