OFFSET
0,2
COMMENTS
The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-right 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+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-2x-x^3) are given by the sequence generated by the row sums.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.205569430400..., 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. 358, 359.
LINKS
FORMULA
T(n,k) = 2^(n - 3k) / ((n - 3k)! k!) * (n - 2k)! where n >= 0 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)/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {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, 18}, {k, 0, Floor[n/3]}] // Flatten
PROG
(GAP) Flat(List([0..20], n->List([0..Int(n/3)], k->2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018
(Magma) /* As triangle */ [[2^(n-3*k)/(Factorial(n-3*k)*Factorial(k))* Factorial(n-2*k): k in [0..Floor(n/3)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018
CROSSREFS
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Jul 30 2018
STATUS
approved