A317494 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 8, 4, 1, 10, 12, 1, 12, 24, 1, 14, 40, 8, 1, 16, 60, 32, 1, 18, 84, 80, 1, 20, 112, 160, 16, 1, 22, 144, 280, 80, 1, 24, 180, 448, 240, 1, 26, 220, 672, 560, 32, 1, 28, 264, 960, 1120, 192, 1, 30, 312, 1320, 2016, 672, 1, 32, 364, 1760, 3360, 1792, 64

Offset

0,5

Comments

The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of 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+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)

The coefficients in the expansion of 1/(1-x-2x^3) are given by the sequence generated by the row sums.

The row sums give A003229.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.695620769559862... (see A289265), 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

Table of n, a(n) for n=0..69.

Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (2 + x)^n

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

Formula

T(n,k) = 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

Example

Triangle begins:
  1;
  1;
  1;
  1,  2;
  1,  4;
  1,  6;
  1,  8,   4;
  1, 10,  12;
  1, 12,  24;
  1, 14,  40,    8;
  1, 16,  60,   32;
  1, 18,  84,   80;
  1, 20, 112,  160,   16;
  1, 22, 144,  280,   80;
  1, 24, 180,  448,  240;
  1, 26, 220,  672,  560,   32;
  1, 28, 264,  960, 1120,  192;
  1, 30, 312, 1320, 2016,  672;
  1, 32, 364, 1760, 3360, 1792, 64;

Mathematica

t[n_, k_] := t[n, k] = 2^k/((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, t[n - 1, k] + 2 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^k/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018

Crossrefs

Row sums give A003229.

Cf. A013609, A038207, A289265, A317495, A128099, A207538.

Sequence in context: A318775 A317500 A339420 * A317505 A137374 A131516

Adjacent sequences: A317491 A317492 A317493 * A317495 A317496 A317497

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 30 2018

Status

approved