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A317503 Triangle read by rows: T(0,0) = 1; T(n,k) = -2 T(n-1,k) + 3 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

%I A317503 #12 Aug 23 2018 16:30:14
%S A317503 1,-2,4,-8,3,16,-12,-32,36,64,-96,9,-128,240,-54,256,-576,216,-512,
%T A317503 1344,-720,27,1024,-3072,2160,-216,-2048,6912,-6048,1080,4096,-15360,
%U A317503 16128,-4320,81,-8192,33792,-41472,15120,-810,16384,-73728,103680,-48384,4860,-32768,159744,-253440,145152,-22680,243
%N A317503 Triangle read by rows: T(0,0) = 1; T(n,k) = -2 T(n-1,k) + 3 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.
%C A317503 The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A303901 ((3-2*x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A317498 ((-2+3x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (3-2*x)^n and (-2+3x)^n are given in A303941 and A302747 respectively.) The coefficients in the expansion of 1/(1 + 2x - 3x^3) are given by the sequence generated by the row sums. The row sums give A317499.
%D A317503 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 136, 396, 397.
%H A317503 Shara Lalo, <a href="/A317503/a317503.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 - 2x)^n</a>
%H A317503 Shara Lalo, <a href="/A317503/a317503_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3x)^n</a>
%F A317503 T(n,k) = (-2)^(n - 3k) * 3^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
%e A317503 Triangle begins:
%e A317503        1;
%e A317503       -2;
%e A317503        4;
%e A317503       -8,       3;
%e A317503       16,     -12;
%e A317503      -32,      36;
%e A317503       64,     -96,       9;
%e A317503     -128,     240,     -54;
%e A317503      256,    -576,     216;
%e A317503     -512,    1344,    -720,      27;
%e A317503     1024,   -3072,    2160,    -216;
%e A317503    -2048,    6912,   -6048,    1080;
%e A317503     4096,  -15360,   16128,   -4320,     81;
%e A317503    -8192,   33792,  -41472,   15120,   -810;
%e A317503    16384,  -73728,  103680,  -48384,   4860;
%e A317503   -32768,  159744, -253440,  145152, -22680,   243;
%e A317503    65536, -344064,  608256, -414720,  90720, -2916;
%t A317503 t[n_, k_] := t[n, k] = (-2)^(n - 3k) * 3^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]  // Flatten
%t A317503 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, -2 * t[n - 1, k] + 3 * t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}] // Flatten
%Y A317503 Row sums give A317499.
%Y A317503 Cf. A303901, A317498.
%Y A317503 Cf. A090388.
%Y A317503 Cf. A303941, A302747.
%K A317503 tabf,sign,easy
%O A317503 0,2
%A A317503 _Shara Lalo_, Aug 02 2018