A317500 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.

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, 1, 32, 312, 960, 560, 1, 34, 364, 1320, 1120, 32

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

Table of n, a(n) for n=0..65.

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

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

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

Row sums give A052942.

Cf. A013609, A038207, A128099, A207538, A256099.

Sequence in context: A327512 A327529 A318775 * A339420 A317494 A317505

Adjacent sequences: A317497 A317498 A317499 * A317501 A317502 A317503

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Sep 03 2018

Status

approved