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%I #14 Jul 23 2023 20:14:15
%S 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,
%T 1308,2415,4,44,270,1072,3200,7884,16954,32960,6,97,838,4353,16126,
%U 47521,119022,264193,534358,9,214,2601,17676,81265,286434,835569,2117656,4815801,10050030
%N 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.
%H G. C. Greubel, <a href="/A117716/b117716.txt">Rows n = 0..50 of the triangle, flattened</a>
%e Triangle begins as:
%e 0;
%e 0, 0;
%e 1, 1, 1;
%e 1, 2, 3, 4;
%e 1, 4, 9, 16, 25;
%e 2, 9, 28, 65, 126, 217;
%e 3, 20, 87, 264, 635, 1308, 2415;
%e 4, 44, 270, 1072, 3200, 7884, 16954, 32960;
%p A117716 := proc(n,m)
%p x^2/(1-(m+1)*x-x^3) ;
%p if n < 0 then
%p 0;
%p else
%p coeftayl(%,x=0,n) ;
%p end if;
%p end proc: # _R. J. Mathar_, May 14 2013
%t T[n_, k_]:= T[n, k]= Coefficient[Series[x^2/(1-(k+1)*x-x^3), {x,0,n+ 2}], x, n];
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten
%o (Magma)
%o m:=12;
%o R<x>:=PowerSeriesRing(Integers(), m+2);
%o A117716:= func< n,k | Coefficient(R!( x^2/(1-(k+1)*x-x^3) ), n) >;
%o [[A117716(n,k): k in [0..n]]: n in [0..m]]; // _G. C. Greubel_, Jul 23 2023
%o (SageMath)
%o def A117716(n,k):
%o P.<x> = PowerSeriesRing(QQ)
%o return P( x^2/(1-(k+1)*x-x^3) ).list()[n]
%o flatten([[A117716(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 23 2023
%Y Cf. A000930 (column 0), A008998 (column 1), A052541 (column 2), A052927 (column 3), A001093 (row 5), A185065 (row 6), A117715, A117724.
%K nonn,tabl
%O 0,8
%A _Roger L. Bagula_, Apr 13 2006, corrected Apr 15 2006
%E Edited by _G. C. Greubel_, Jul 23 2023
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