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%I #15 Sep 05 2018 02:26:28
%S 1,-1,1,2,-1,-4,1,6,4,-1,-8,-12,1,10,24,8,-1,-12,-40,-32,1,14,60,80,
%T 16,-1,-16,-84,-160,-80,1,18,112,280,240,32,-1,-20,-144,-448,-560,
%U -192,1,22,180,672,1120,672,64,-1,-24,-220,-960,-2016,-1792,-448
%N Triangle read by rows: T(0,0) = 1; T(n,k) = -T(n-1,k) + 2 T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.
%C The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2*x)^n).
%C The coefficients in the expansion of 1/(1+x-2x^2) are given by the sequence generated by the row sums.
%C When n is even the numbers in the row are positive, and when n is odd the numbers in the row are negative.
%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.
%F G.f.: 1 / (1 + t*x - 2t^2).
%e Triangle begins:
%e 1;
%e -1;
%e 1, 2;
%e -1, -4;
%e 1, 6, 4;
%e -1, -8, -12;
%e 1, 10, 24, 8;
%e -1, -12, -40, -32;
%e 1, 14, 60, 80, 16;
%e -1, -16, -84, -160, -80;
%e 1, 18, 112, 280, 240, 32;
%e -1, -20, -144, -448, -560, -192;
%e 1, 22, 180, 672, 1120, 672, 64;
%e -1, -24, -220, -960, -2016, -1792, -448;
%e 1, 26, 264, 1320, 3360, 4032, 1792, 128;
%e -1, -28, -312, -1760, -5280, -8064, -5376, -1024;
%e 1, 30, 364, 2288, 7920, 14784, 13440, 4608, 256;
%e -1, -32, -420, -2912, -11440, -25344, -29568, -15360, -2304;
%t t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, -t[n - 1, k] + 2 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
%o (PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, -T(n-1, k) + 2*T(n-2, k-1)));
%o tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 26 2018
%Y Signed version of A128099.
%Y Row sums give A077925.
%Y Cf. A303872, A033999 (column 0).
%K tabf,easy,sign
%O 0,4
%A _Shara Lalo_, May 25 2018
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