cp's OEIS Frontend

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.

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