OFFSET
0,2
COMMENTS
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.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 136, 396, 397.
LINKS
FORMULA
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).
EXAMPLE
Triangle begins:
1;
3;
9;
27, -2;
81, -12;
243, -54;
729, -216, 4;
2187, -810, 36;
6561, -2916, 216;
19683, -10206, 1080, -8;
59049, -34992, 4860, -96;
177147, -118098, 20412, -720;
531441, -393660, 81648, -4320, 16;
1594323, -1299078, 314928, -22680, 240;
4782969, -4251528, 1180980, -108864, 2160;
14348907, -13817466, 4330260, -489888, 15120, -32;
43046721, -44641044, 15588936, -2099520, 90720, -576;
MATHEMATICA
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[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
CROSSREFS
KEYWORD
tabf,sign,easy
AUTHOR
Shara Lalo, Aug 02 2018
STATUS
approved