A141604 Triangle, read by rows, T(n,k) = round(A006720(n)/(A006720(n-k)*A006720(k))).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 4, 7, 4, 2, 1, 1, 3, 8, 12, 12, 8, 3, 1, 1, 3, 8, 20, 15, 20, 8, 3, 1, 1, 5, 14, 45, 52, 52, 45, 14, 5, 1, 1, 5, 26, 66, 109, 170, 109, 66, 26, 5, 1

Offset: 0

Author

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

Keywords

• nonn
• less
• tabl

Comments

The round-function in the definition is round-to-nearest, not Mathematica's round-to-even. - R. J. Mathar, Jul 12 2012

Examples

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   1,   1,   1;
  1,   2,   2,   2,   1;
  1,   2,   3,   3,   2,   1;
  1,   2,   4,   7,   4,   2,   1;
  1,   3,   8,  12,  12,   8,   3,   1;
  1,   3,   8,  20,  15,  20,   8,   3,   1;
  1,   5,  14,  45,  52,  52,  45,  14,   5,   1;
  1,   5,  26,  66, 109, 170, 109,  66,  26,   5,   1;
		

Links

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

Crossrefs

Cf. A006720.

Programs

Magma

A006720:= [n le 4 select 1 else (Self(n-1)*Self(n-3)+Self(n-2)^2)/Self(n-4): n in [1..30]];
A141604:= func< n,k | Round(A006720[n+1]/(A006720[k+1]*A006720[n-k+1])) >;
[A141604(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 21 2024

Maple

A141604 := proc(n,m)
        round(A006720(n)/A006720(n-m)/A006720(m)) ;
end proc:
seq(seq(A141604(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 12 2012

Mathematica

f[n_]:= f[n]= If[n<4, 1, (f[n-1]*f[n-3] +f[n-2]^2)/f[n-4]]; (* A006720 *)
A141604[n_, k_]:= Round[f[n]/(f[k]*f[n-k])];
Table[A141604[n,k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Sep 21 2024 *)

SageMath

def f(n): # f = A006720
    if n<4: return 1
    else: return (f(n-1)*f(n-3) +f(n-2)^2)/f(n-4)
def A141604(n,k): return round(f(n)/(f(k)*f(n-k)))
flatten([[A141604(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 21 2024