A317026 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 8 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

1, 1, 1, 8, 1, 16, 1, 24, 64, 1, 32, 192, 1, 40, 384, 512, 1, 48, 640, 2048, 1, 56, 960, 5120, 4096, 1, 64, 1344, 10240, 20480, 1, 72, 1792, 17920, 61440, 32768, 1, 80, 2304, 28672, 143360, 196608, 1, 88, 2880, 43008, 286720, 688128, 262144, 1, 96, 3520, 61440, 516096, 1835008, 1835008

Offset: 0

Author

Zagros Lalo, Jul 19 2018

Keywords

• tabf
• nonn
• easy

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013615 ((1+8*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A038279 ((8+x)^n).

The coefficients in the expansion of 1/(1-x-8*x^2) are given by the sequence generated by the row sums.

The row sums are Generalized Fibonacci numbers (see A015443).

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.3722813232690143..., when n approaches infinity; see A235162 (Decimal expansion of (sqrt(33)+1)/2).

Examples

			Triangle begins:
  1;
  1;
  1, 8;
  1, 16;
  1, 24, 64;
  1, 32, 192;
  1, 40, 384, 512;
  1, 48, 640, 2048;
  1, 56, 960, 5120, 4096;
  1, 64, 1344, 10240, 20480;
  1, 72, 1792, 17920, 61440, 32768;
  1, 80, 2304, 28672, 143360, 196608;
  1, 88, 2880, 43008, 286720, 688128, 262144;
  1, 96, 3520, 61440, 516096, 1835008, 1835008;
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 70, 98

Links

• Zagros Lalo, Left-justified triangle
• Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 8x)^n
• Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (8 + x)^n

Crossrefs

Row sums give A015443.

Cf. A013615, A038279, A235162.

Programs

GAP

Flat(List([0..13],n->List([0..Int(n/2)],k->8^k*Binomial(n-k,k)))); # Muniru A Asiru, Jul 19 2018

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 8 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
Table[8^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

Formula

T(n,k) = 8^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).