A318776 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) + T(n-5,k-1) for k = 0..floor(n/5); T(n,k)=0 for n or k < 0.

1, 2, 4, 8, 16, 32, 1, 64, 4, 128, 12, 256, 32, 512, 80, 1024, 192, 1, 2048, 448, 6, 4096, 1024, 24, 8192, 2304, 80, 16384, 5120, 240, 32768, 11264, 672, 1, 65536, 24576, 1792, 8, 131072, 53248, 4608, 40, 262144, 114688, 11520, 160, 524288, 245760, 28160, 560, 1048576, 524288, 67584, 1792, 1, 2097152, 1114112, 159744, 5376, 10

Offset

0,2

Comments

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

The coefficients in the expansion of 1/(1-2x-x^5) are given by the sequence generated by the row sums.

The row sums give A098588.

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 2.0559673967128..., 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..59.

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

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

Formula

T(n,k) = 2^(n - 5*k) / ((n - 5*k)! k!) * (n - 4*k)! where n >= 0 and 0 <= k <= floor(n/5).

Example

Triangle begins:
        1;
        2;
        4;
        8;
       16;
       32,       1;
       64,       4;
      128,      12;
      256,      32;
      512,      80;
     1024,     192,      1;
     2048,     448,      6;
     4096,    1024,     24;
     8192,    2304,     80;
    16384,    5120,    240;
    32768,   11264,    672,    1;
    65536,   24576,   1792,    8;
   131072,   53248,   4608,   40;
   262144,  114688,  11520,  160;
   524288,  245760,  28160,  560;
  1048576,  524288,  67584, 1792,  1;
  2097152, 1114112, 159744, 5376, 10;
  ...

Mathematica

t[n_, k_] := t[n, k] = 2^(n - 5 k)/((n - 5 k)! k!) (n - 4 k)!; Table[t[n, k], {n, 0, 22}, {k, 0, Floor[n/5]} ]  // Flatten.
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 t[n - 1, k] + t[n - 5, k - 1]]; Table[t[n, k], {n, 0, 22}, {k, 0, Floor[n/5]}] // Flatten.

Crossrefs

Row sums give A098588.

Cf. A013609, A038207, A128099, A207538.

Cf. also A000079 (column 0), A001787 (column 1), A001788 (column 2), A001789 (column 3)

Sequence in context: A243083 A239561 A010747 * A036130 A335836 A122169

Adjacent sequences: A318773 A318774 A318775 * A318777 A318778 A318779

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Sep 04 2018

Status

approved