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%I #13 Jul 18 2023 04:12:00
%S -1,-2,1,-2,2,-1,-4,4,-2,1,-4,8,-6,2,-1,-8,12,-12,8,-2,1,-8,24,-24,16,
%T -10,2,-1,-16,32,-48,40,-20,12,-2,1,-16,64,-80,80,-60,24,-14,2,-1,-32,
%U 80,-160,160,-120,84,-28,16,-2,1,-32,160,-240,320,-280,168,-112,32,-18,2,-1
%N Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x) = -1, p(1,x) = x-2, p(n,x) = -x*p(n-1,x) + 2*p(n-2,x) for n >= 3 and 0 <= k <= n.
%H G. C. Greubel, <a href="/A123148/b123148.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = [x^k]( p(n,x) ), where p(0,x) = -1, p(1,x) = x-2, p(n,x) = -x*p(n-1,x) + 2*p(n-2,x).
%F Sum_{k=0..n} T(n, k) = -1.
%F Sum_{k=0..n} (-1)^k * T(n,k) = -A001045(n+2).
%F From _G. C. Greubel_, Jul 17 2023: (Start)
%F T(n,k) = (-1)^(k+1)*2^Floor((n-k+1)/2)*Binomial( Floor((n+k)/2), k).
%F T(n,k) = (-1)^(k+1)*2^Floor((n-k+1)/2)*A046854(n,k).
%F T(n,0) = -A016116(n+1).
%F T(n,1) = A171647(n).
%F Sum_{k=0..n} (-1)^k * abs(T(n,k)) = 1.
%F Sum_{k=0..floor(n/2)} T(n-k,k) = - A000034(n).
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = -A038754(n). (End)
%e The first few polynomials, p(n,x), are:
%e p(0,x) = -1;
%e p(1,x) = -2 + x;
%e p(2,x) = -2 + 2*x - x^2;
%e p(3,x) = -4 + 4*x - 2*x^2 + x^3;
%e p(4,x) = -4 + 8*x - 6*x^2 + 2*x^3 - x^4;
%e The triangle, T(n, k) = [x^k] p(n, x), begins as:
%e -1;
%e -2, 1;
%e -2, 2, -1;
%e -4, 4, -2, 1;
%e -4, 8, -6, 2, -1;
%e -8, 12, -12, 8, -2, 1;
%e -8, 24, -24, 16, -10, 2, -1;
%p p[0]:=-1: p[1]:=x-2: for n from 2 to 10 do p[n]:=sort(expand(-x*p[n-1]+2*p[n-2])) od: for n from 0 to 10 do seq(coeff(p[n],x,k),k=0..n) od; # yields sequence in triangular form
%t p[0,x]= -1; p[1,x]= x-2; p[k_, x_]:= p[k,x]= -x*p[k-1,x] + 2*p[k-2,x];
%t T[n_, k_]:= Coefficient[p[n, x], x, k];
%t Table[T[n,k], {n,0,12},{k,0,n}]//Flatten
%o (Magma)
%o A123148:=func< n,k | (-1)^(k+1)*2^Floor((n-k+1)/2)*Binomial( Floor((n+k)/2), k) >;
%o [A123148(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 17 2023
%o (SageMath)
%o def A123148(n,k): return (-1)^(k+1)*2^((n-k+1)//2)*binomial((n+k)//2, k)
%o flatten([[A123148(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 17 2023
%Y Cf. A000034, A001045, A016116, A038754, A046854, A171647.
%K sign,tabl
%O 0,2
%A _Roger L. Bagula_, Oct 01 2006
%E Edited by _N. J. A. Sloane_, Oct 29 2006
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