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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.

%I A117724 #12 Jul 24 2023 01:24:49
%S A117724 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,
%T A117724 10,12,14,16,2,9,28,65,126,217,344,513,730,3,12,27,48,75,108,147,192,
%U A117724 243,300,4,22,90,268,640,1314,2422,4120,6588,10030,14674
%N 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.
%H A117724 Nathaniel Johnston, <a href="/A117724/b117724.txt">Rows n = 0..50, flattened</a>
%F A117724 T(n,k) = coefficient [x^n] ( x^2/(1-(k+1)*x^2-x^3) ).
%F A117724 T(n, 0) = A000931(n+1).
%F A117724 T(n, 1) = A008346(n-2) = (-1)^(n-1)*A119282(n-1).
%F A117724 T(n, 2) = A052931(n-2).
%e A117724 The table starts:
%e A117724   0;
%e A117724   0,  0;
%e A117724   1,  1,  1;
%e A117724   0,  0,  0,  0;
%e A117724   1,  2,  3,  4,   5;
%e A117724   1,  1,  1,  1,   1,   1;
%e A117724   1,  4,  9, 16,  25,  36,  49;
%e A117724   2,  4,  6,  8,  10,  12,  14,  16;
%e A117724   2,  9, 28, 65, 126, 217, 344, 513, 730;
%e A117724   3, 12, 27, 48,  75, 108, 147, 192, 243, 300;
%p A117724 t:=taylor(x^2/(1-(k+1)*x^2-x^3), x, 15):
%p A117724 seq(seq(coeff(t,x,n), k=0..n),n=0..12); # _Nathaniel Johnston_, Apr 27 2011
%t A117724 T[n_, k_]:= T[n, k]= Coefficient[Series[x^2/(1-(k+1)*x^2-x^3), {x,0,n+ 2}], x, n];
%t A117724 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o A117724 (Magma)
%o A117724 m:=12;
%o A117724 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A117724 A117724:= func< n, k | Coefficient(R!( x^2/(1-(k+1)*x^2-x^3) ), n) >;
%o A117724 [A117724(n, k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Jul 23 2023
%o A117724 (SageMath)
%o A117724 def A117724(n, k):
%o A117724     P.<x> = PowerSeriesRing(QQ)
%o A117724     return P( x^2/(1-(k+1)*x^2-x^3) ).list()[n]
%o A117724 flatten([[A117724(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 23 2023
%Y A117724 Cf. A000931, A008346, A052931, A117716, A119282.
%K A117724 nonn,tabl,easy
%O A117724 0,12
%A A117724 _Roger L. Bagula_, Apr 13 2006
%E A117724 Sign in definition corrected, offset set to -1 by Assoc. Eds. of the OEIS, Jun 15 2010
%E A117724 Edited by _G. C. Greubel_, Jul 23 2023