A020708 -id:A020708 - cp's OEIS frontend

A317495 Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

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

Offset: 0

Zagros Lalo, Jul 30 2018

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.
The row sums give A008998 and Pisot sequences E(4,9), P(4,9) when n > 1, see A020708.
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.

			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;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359.

Row sums give A008998, A020708.
Cf. A013609
Cf. A038207
Cf. A317494
Cf. A128099, A207538.
Cf. A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3), A003472 (column 4).

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

/* 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

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

T(n,k) = 2^(n - 3k) / ((n - 3k)! k!) * (n - 2k)! where n >= 0 and k = 0..floor(n/3).

cp's OEIS Frontend

A317495 Triangle read by rows: T(0,0) = 1; T(n,k) =2 * T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

Formula