OFFSET
0,6
COMMENTS
The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-x-2x^4) are given by the sequence generated by the row sums.
The row sums give A052942.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.543689012692076... (A256099: Decimal expansion of the real root of a cubic used by Omar Khayyám in a geometrical problem), 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^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).
EXAMPLE
Triangle begins:
1;
1;
1;
1;
1, 2;
1, 4;
1, 6;
1, 8;
1, 10, 4;
1, 12, 12;
1, 14, 24;
1, 16, 40;
1, 18, 60, 8;
1, 20, 84, 32;
1, 22, 112, 80;
1, 24, 144, 160;
1, 26, 180, 280, 16;
1, 28, 220, 448, 80;
1, 30, 264, 672, 240;
...
MATHEMATICA
t[n_, k_] := t[n, k] = 2^k/((n - 4 k)! k!) (n - 3 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/4]} ] // Flatten.
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/4]}] // Flatten.
CROSSREFS
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Sep 03 2018
STATUS
approved