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A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

%I A317505 #8 Aug 23 2018 16:30:39
%S A317505 1,-1,1,-1,2,1,-4,-1,6,1,-8,4,-1,10,-12,1,-12,24,-1,14,-40,8,1,-16,60,
%T A317505 -32,-1,18,-84,80,1,-20,112,-160,16,-1,22,-144,280,-80,1,-24,180,-448,
%U A317505 240,-1,26,-220,672,-560,32,1,-28,264,-960,1120,-192,-1,30,-312,1320,-2016,672,1,-32,364,-1760,3360,-1792,64,-1,34,-420,2288,-5280,4032,-448
%N A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
%C A317505 The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1+x+2x^3) are given by the sequence generated by the row sums (see A077973).
%D A317505 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.
%H A317505 Shara Lalo, <a href="/A317505/a317505.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n</a>
%H A317505 Shara Lalo, <a href="/A317505/a317505_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n</a>
%F A317505 T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
%e A317505 Triangle begins:
%e A317505    1;
%e A317505   -1;
%e A317505    1;
%e A317505   -1,   2;
%e A317505    1,  -4;
%e A317505   -1,   6;
%e A317505    1,  -8,    4;
%e A317505   -1,  10,  -12;
%e A317505    1, -12,   24;
%e A317505   -1,  14,  -40,     8;
%e A317505    1, -16,   60,   -32;
%e A317505   -1,  18,  -84,    80;
%e A317505    1, -20,  112,  -160,    16;
%e A317505   -1,  22, -144,   280,   -80;
%e A317505    1, -24,  180,  -448,   240;
%e A317505   -1,  26, -220,   672,  -560,    32;
%e A317505    1, -28,  264,  -960,  1120,  -192;
%e A317505   -1,  30, -312,  1320, -2016,   672;
%e A317505    1, -32,  364, -1760,  3360, -1792,   64;
%e A317505   -1,  34, -420,  2288, -5280,  4032, -448;
%t A317505 t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ]  // Flatten
%t A317505 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten
%Y A317505 Row sums give A077973.
%Y A317505 Cf. A065109, A303872.
%Y A317505 Cf. A133156, A305098.
%K A317505 tabf,sign,easy
%O A317505 0,5
%A A317505 _Shara Lalo_, Aug 02 2018