A117724 Triangle T(n,k) = coefficient [x^n] of x^2/(1-(k+1)*x^2-x^3) for row n, and columns k = 0..n, read by rows.

0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 4, 9, 16, 25, 36, 49, 2, 4, 6, 8, 10, 12, 14, 16, 2, 9, 28, 65, 126, 217, 344, 513, 730, 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 4, 22, 90, 268, 640, 1314, 2422, 4120, 6588, 10030, 14674

Offset: 0

Author

Roger L. Bagula, Apr 13 2006

Keywords

• nonn
• tabl
• easy

Examples

			The table starts:
  0;
  0,  0;
  1,  1,  1;
  0,  0,  0,  0;
  1,  2,  3,  4,   5;
  1,  1,  1,  1,   1,   1;
  1,  4,  9, 16,  25,  36,  49;
  2,  4,  6,  8,  10,  12,  14,  16;
  2,  9, 28, 65, 126, 217, 344, 513, 730;
  3, 12, 27, 48,  75, 108, 147, 192, 243, 300;
		

Links

• Nathaniel Johnston, Rows n = 0..50, flattened

Crossrefs

Cf. A000931, A008346, A052931, A117716, A119282.

Programs

Magma

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A117724:= func< n, k | Coefficient(R!( x^2/(1-(k+1)*x^2-x^3) ), n) >;
[A117724(n, k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 23 2023

Maple

t:=taylor(x^2/(1-(k+1)*x^2-x^3), x, 15):
seq(seq(coeff(t,x,n), k=0..n),n=0..12); # Nathaniel Johnston, Apr 27 2011

Mathematica

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

SageMath

def A117724(n, k):
    P. = PowerSeriesRing(QQ)
    return P( x^2/(1-(k+1)*x^2-x^3) ).list()[n]
flatten([[A117724(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 23 2023

Formula

T(n,k) = coefficient [x^n] ( x^2/(1-(k+1)*x^2-x^3) ).

T(n, 0) = A000931(n+1).

T(n, 1) = A008346(n-2) = (-1)^(n-1)*A119282(n-1).

T(n, 2) = A052931(n-2).

Extensions

Sign in definition corrected, offset set to -1 by Assoc. Eds. of the OEIS, Jun 15 2010

Edited by G. C. Greubel, Jul 23 2023