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
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
Programs
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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
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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