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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. 2
1, -2, 4, -8, 3, 16, -12, -32, 36, 64, -96, 9, -128, 240, -54, 256, -576, 216, -512, 1344, -720, 27, 1024, -3072, 2160, -216, -2048, 6912, -6048, 1080, 4096, -15360, 16128, -4320, 81, -8192, 33792, -41472, 15120, -810, 16384, -73728, 103680, -48384, 4860, -32768, 159744, -253440, 145152, -22680, 243 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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

Table of n, a(n) for n=0..50.

Shara Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 - 2x)^n

Shara Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3x)^n

FORMULA

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).

EXAMPLE

Triangle begins:

       1;

      -2;

       4;

      -8,       3;

      16,     -12;

     -32,      36;

      64,     -96,       9;

    -128,     240,     -54;

     256,    -576,     216;

    -512,    1344,    -720,      27;

    1024,   -3072,    2160,    -216;

   -2048,    6912,   -6048,    1080;

    4096,  -15360,   16128,   -4320,     81;

   -8192,   33792,  -41472,   15120,   -810;

   16384,  -73728,  103680,  -48384,   4860;

  -32768,  159744, -253440,  145152, -22680,   243;

   65536, -344064,  608256, -414720,  90720, -2916;

MATHEMATICA

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[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

CROSSREFS

Row sums give A317499.

Cf. A303901, A317498.

Cf. A090388.

Cf. A303941, A302747.

Sequence in context: A242365 A119436 A277695 * A243505 A243065 A289271

Adjacent sequences:  A317500 A317501 A317502 * A317504 A317505 A317506

KEYWORD

tabf,sign,easy

AUTHOR

Shara Lalo, Aug 02 2018

STATUS

approved

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Last modified December 5 15:11 EST 2019. Contains 329753 sequences. (Running on oeis4.)