A317506 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

1, 2, 4, 8, 16, -1, 32, -4, 64, -12, 128, -32, 256, -80, 1, 512, -192, 6, 1024, -448, 24, 2048, -1024, 80, 4096, -2304, 240, -1, 8192, -5120, 672, -8, 16384, -11264, 1792, -40, 32768, -24576, 4608, -160, 65536, -53248, 11520, -560, 1, 131072, -114688, 28160, -1792, 10

Offset: 0

Author

Shara Lalo, Aug 31 2018

Keywords

• tabf
• sign
• easy

Comments

The numbers in rows of the triangle are along "third layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "third 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^4) are given by the sequence generated by the row sums. The row sums give A008937. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.83928675521416113... (A058265: Decimal expansion of the tribonacci constant t, the real root of x^3-x^2-x-1), when n approaches infinity.

Examples

			Triangle begins:
       1;
       2;
       4;
       8;
      16,      -1;
      32,      -4;
      64,     -12;
     128,     -32;
     256,     -80,     1;
     512,    -192,     6;
    1024,    -448,    24;
    2048,   -1024,    80;
    4096,   -2304,   240,    -1;
    8192,   -5120,   672,    -8;
   16384,  -11264,  1792,   -40;
   32768,  -24576,  4608,  -160;
   65536,  -53248, 11520,  -560,  1;
  131072, -114688, 28160, -1792, 10;
  262144, -245760, 67584, -5376, 60;
		

References

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

Links

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

Crossrefs

Row sums give A008937.

Cf. A065109, A303872.

Cf. A133156, A305098.

Cf. A058265.

Programs

Mathematica

t[n_, k_] := t[n, k] = 2^(n - 4 k) * (-1)^k/((n - 4 k)! k!) * (n - 3 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}] // Flatten

Formula

T(n,k) = 2^(n - 4*k) * (-1)^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).