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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.
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%I #8 Aug 23 2018 16:30:02

%S 1,3,9,27,-2,81,-12,243,-54,729,-216,4,2187,-810,36,6561,-2916,216,

%T 19683,-10206,1080,-8,59049,-34992,4860,-96,177147,-118098,20412,-720,

%U 531441,-393660,81648,-4320,16,1594323,-1299078,314928,-22680,240,4782969,-4251528,1180980,-108864,2160

%N 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 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 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 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 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 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 Triangle begins:

%e 1;

%e 3;

%e 9;

%e 27, -2;

%e 81, -12;

%e 243, -54;

%e 729, -216, 4;

%e 2187, -810, 36;

%e 6561, -2916, 216;

%e 19683, -10206, 1080, -8;

%e 59049, -34992, 4860, -96;

%e 177147, -118098, 20412, -720;

%e 531441, -393660, 81648, -4320, 16;

%e 1594323, -1299078, 314928, -22680, 240;

%e 4782969, -4251528, 1180980, -108864, 2160;

%e 14348907, -13817466, 4330260, -489888, 15120, -32;

%e 43046721, -44641044, 15588936, -2099520, 90720, -576;

%t 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 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 Row sums give A077846.

%Y Cf. A303901, A317498.

%Y Cf. A090388.

%Y Cf. A303941, A302747.

%K tabf,sign,easy

%O 0,2

%A _Shara Lalo_, Aug 02 2018