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A317502 Triangle read by rows: T(0,0) = 1; T(n,k) = 3 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 A317502 #8 Aug 23 2018 16:30:02
%S A317502 1,3,9,27,-2,81,-12,243,-54,729,-216,4,2187,-810,36,6561,-2916,216,
%T A317502 19683,-10206,1080,-8,59049,-34992,4860,-96,177147,-118098,20412,-720,
%U A317502 531441,-393660,81648,-4320,16,1594323,-1299078,314928,-22680,240,4782969,-4251528,1180980,-108864,2160
%N A317502 Triangle read by rows: T(0,0) = 1; T(n,k) = 3 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 A317502 The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303901 ((3-2*x)^n) and along "second layer" skew diagonals pointing top-left 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/(1-3x+2x^3) are given by the sequence generated by the row sums. The row sums give A077846. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.7320508075688772... (A090388: 1+sqrt(3)), when n approaches infinity.
%D A317502 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 A317502 Shara Lalo, <a href="/A317502/a317502.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 - 2x)^n</a>
%H A317502 Shara Lalo, <a href="/A317502/a317502_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3x)^n</a>
%F A317502 T(n,k) = 3^(n - 3k) * (-2)^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).
%e A317502 Triangle begins:
%e A317502         1;
%e A317502         3;
%e A317502         9;
%e A317502         27,        -2;
%e A317502         81,       -12;
%e A317502        243,       -54;
%e A317502        729,      -216,        4;
%e A317502       2187,      -810,       36;
%e A317502       6561,     -2916,      216;
%e A317502      19683,    -10206,     1080,       -8;
%e A317502      59049,    -34992,     4860,      -96;
%e A317502     177147,   -118098,    20412,     -720;
%e A317502     531441,   -393660,    81648,    -4320,    16;
%e A317502    1594323,  -1299078,   314928,   -22680,   240;
%e A317502    4782969,  -4251528,  1180980,  -108864,  2160;
%e A317502   14348907, -13817466,  4330260,  -489888, 15120,  -32;
%e A317502   43046721, -44641044, 15588936, -2099520, 90720, -576;
%t A317502 t[n_, k_] := t[n, k] = 3^(n - 3k) * (-2)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]  // Flatten
%t A317502 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 * t[n - 1, k] - 2 * t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}] // Flatten
%Y A317502 Row sums give A077846.
%Y A317502 Cf. A303901, A317498.
%Y A317502 Cf. A090388.
%Y A317502 Cf. A303941, A302747.
%K A317502 tabf,sign,easy
%O A317502 0,2
%A A317502 _Shara Lalo_, Aug 02 2018