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A305098 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.

%I A305098 #15 Sep 05 2018 02:26:28
%S A305098 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 A305098 16,-1,-16,-84,-160,-80,1,18,112,280,240,32,-1,-20,-144,-448,-560,
%U A305098 -192,1,22,180,672,1120,672,64,-1,-24,-220,-960,-2016,-1792,-448
%N A305098 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 A305098 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 A305098 The coefficients in the expansion of 1/(1+x-2x^2) are given by the sequence generated by the row sums.
%C A305098 When n is even the numbers in the row are positive, and when n is odd the numbers in the row are negative.
%D A305098 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 389-391.
%F A305098 G.f.: 1 / (1 + t*x - 2t^2).
%e A305098 Triangle begins:
%e A305098    1;
%e A305098   -1;
%e A305098    1,   2;
%e A305098   -1,  -4;
%e A305098    1,   6,    4;
%e A305098   -1,  -8,  -12;
%e A305098    1,  10,   24,     8;
%e A305098   -1, -12,  -40,   -32;
%e A305098    1,  14,   60,    80,     16;
%e A305098   -1, -16,  -84,  -160,    -80;
%e A305098    1,  18,  112,   280,    240,     32;
%e A305098   -1, -20, -144,  -448,   -560,   -192;
%e A305098    1,  22,  180,   672,   1120,    672,     64;
%e A305098   -1, -24, -220,  -960,  -2016,  -1792,   -448;
%e A305098    1,  26,  264,  1320,   3360,   4032,   1792,    128;
%e A305098   -1, -28, -312, -1760,  -5280,  -8064,  -5376,  -1024;
%e A305098    1,  30,  364,  2288,   7920,  14784,  13440,   4608,   256;
%e A305098   -1, -32, -420, -2912, -11440, -25344, -29568, -15360, -2304;
%t A305098 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 A305098 (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 A305098 tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 26 2018
%Y A305098 Signed version of A128099.
%Y A305098 Row sums give A077925.
%Y A305098 Cf. A303872, A033999 (column 0).
%K A305098 tabf,easy,sign
%O A305098 0,4
%A A305098 _Shara Lalo_, May 25 2018