A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - 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.

1, -1, 1, -1, 2, 1, -4, -1, 6, 1, -8, 4, -1, 10, -12, 1, -12, 24, -1, 14, -40, 8, 1, -16, 60, -32, -1, 18, -84, 80, 1, -20, 112, -160, 16, -1, 22, -144, 280, -80, 1, -24, 180, -448, 240, -1, 26, -220, 672, -560, 32, 1, -28, 264, -960, 1120, -192, -1, 30, -312, 1320, -2016, 672, 1, -32, 364, -1760, 3360, -1792, 64, -1, 34, -420, 2288, -5280, 4032, -448

Offset: 0

Author

Shara Lalo, Aug 02 2018

Keywords

• tabf
• sign
• easy

Comments

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1+x+2x^3) are given by the sequence generated by the row sums (see A077973).

Examples

			Triangle begins:
   1;
  -1;
   1;
  -1,   2;
   1,  -4;
  -1,   6;
   1,  -8,    4;
  -1,  10,  -12;
   1, -12,   24;
  -1,  14,  -40,     8;
   1, -16,   60,   -32;
  -1,  18,  -84,    80;
   1, -20,  112,  -160,    16;
  -1,  22, -144,   280,   -80;
   1, -24,  180,  -448,   240;
  -1,  26, -220,   672,  -560,    32;
   1, -28,  264,  -960,  1120,  -192;
  -1,  30, -312,  1320, -2016,   672;
   1, -32,  364, -1760,  3360, -1792,   64;
  -1,  34, -420,  2288, -5280,  4032, -448;
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Links

• Shara Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n
• Shara Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n

Crossrefs

Row sums give A077973.

Cf. A065109, A303872.

Cf. A133156, A305098.

Programs

Mathematica

t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten

Formula

T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).