A317504 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

1, 2, 4, 8, -1, 16, -4, 32, -12, 64, -32, 1, 128, -80, 6, 256, -192, 24, 512, -448, 80, -1, 1024, -1024, 240, -8, 2048, -2304, 672, -40, 4096, -5120, 1792, -160, 1, 8192, -11264, 4608, -560, 10, 16384, -24576, 11520, -1792, 60, 32768, -53248, 28160, -5376, 280, -1, 65536, -114688, 67584, -15360, 1120, -12

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 A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-left 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/(1-2x+x^3) are given by the sequence generated by the row sums. The row sums give A000071 (Fibonacci numbers - 1). If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.61803398874989484... (A001622: Decimal expansion of Golden ratio (phi or tau) = (1 + sqrt(5))/2), 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. 139-141, 391-393.

Links

Table of n, a(n) for n=0..56.

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

Formula

T(n,k) = 2^(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;
       2;
       4;
       8,      -1;
      16,      -4;
      32,     -12;
      64,     -32,      1;
     128,     -80,      6;
     256,    -192,     24;
     512,    -448,     80,      -1;
    1024,   -1024,    240,      -8;
    2048,   -2304,    672,     -40;
    4096,   -5120,   1792,    -160,     1;
    8192,  -11264,   4608,    -560,    10;
   16384,  -24576,  11520,   -1792,    60;
   32768,  -53248,  28160,   -5376,   280,   -1;
   65536, -114688,  67584,  -15360,  1120,  -12;
  131072, -245760, 159744,  -42240,  4032,  -84;
  262144, -524288, 372736, -112640, 13440, -448, 1;

Mathematica

t[n_, k_] := t[n, k] = 2^(n - 3k) * (-1)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 16}, {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] - t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/3]}] // Flatten

Crossrefs

Row sums give A000071.

Cf. A065109, A303872.

Cf. A133156, A305098.

Cf. A001622.

Sequence in context: A000455 A282821 A317495 * A097888 A030275 A097874

Adjacent sequences: A317501 A317502 A317503 * A317505 A317506 A317507

Keyword

tabf,sign,easy

Author

Shara Lalo, Aug 02 2018

Status

approved