OFFSET
0,2
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.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.
LINKS
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).
EXAMPLE
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;
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
CROSSREFS
KEYWORD
tabf,sign,easy
AUTHOR
Shara Lalo, Aug 31 2018
STATUS
approved