A117716 Triangle T(n,k) read by rows: the coefficient [x^n] of x^2/(1-(k+1)*x-x^3) in row n, columns 0 <= k <= n.

0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 1, 4, 9, 16, 25, 2, 9, 28, 65, 126, 217, 3, 20, 87, 264, 635, 1308, 2415, 4, 44, 270, 1072, 3200, 7884, 16954, 32960, 6, 97, 838, 4353, 16126, 47521, 119022, 264193, 534358, 9, 214, 2601, 17676, 81265, 286434, 835569, 2117656, 4815801, 10050030

Offset: 0

Author

Roger L. Bagula, Apr 13 2006, corrected Apr 15 2006

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  0;
  0,  0;
  1,  1,   1;
  1,  2,   3,    4;
  1,  4,   9,   16,   25;
  2,  9,  28,   65,  126,  217;
  3, 20,  87,  264,  635, 1308,  2415;
  4, 44, 270, 1072, 3200, 7884, 16954, 32960;
		

Links

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

Crossrefs

Cf. A000930 (column 0), A008998 (column 1), A052541 (column 2), A052927 (column 3), A001093 (row 5), A185065 (row 6), A117715, A117724.

Programs

Magma

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A117716:= func< n,k | Coefficient(R!( x^2/(1-(k+1)*x-x^3) ), n) >;
[[A117716(n,k): k in [0..n]]: n in [0..m]]; // G. C. Greubel, Jul 23 2023

Maple

A117716 := proc(n,m)
        x^2/(1-(m+1)*x-x^3) ;
        if n < 0 then
                0;
        else
                coeftayl(%,x=0,n) ;
        end if;
end proc: # R. J. Mathar, May 14 2013

Mathematica

T[n_, k_]:= T[n, k]= Coefficient[Series[x^2/(1-(k+1)*x-x^3), {x,0,n+ 2}], x, n];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

SageMath

def A117716(n,k):
    P. = PowerSeriesRing(QQ)
    return P( x^2/(1-(k+1)*x-x^3) ).list()[n]
flatten([[A117716(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 23 2023

Extensions

Edited by G. C. Greubel, Jul 23 2023