A317016 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 7 * T(n-2,k-1) for k = 0..floor(n/2). T(n,k)=0 for n or k < 0.

1, 1, 1, 7, 1, 14, 1, 21, 49, 1, 28, 147, 1, 35, 294, 343, 1, 42, 490, 1372, 1, 49, 735, 3430, 2401, 1, 56, 1029, 6860, 12005, 1, 63, 1372, 12005, 36015, 16807, 1, 70, 1764, 19208, 84035, 100842, 1, 77, 2205, 28812, 168070, 352947, 117649, 1, 84, 2695, 41160, 302526, 941192, 823543

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 A013614 ((1+7*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A027466 ((7+x)^n).

The coefficients in the expansion of 1/(1-x-7*x^2) are given by the sequence generated by the row sums.

The row sums are Generalized Fibonacci numbers (see A015442).

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.192582403567252..., when n approaches infinity (see A223140).

Examples

			Triangle begins:
  1;
  1;
  1,  7;
  1, 14;
  1, 21,   49;
  1, 28,  147;
  1, 35,  294,   343;
  1, 42,  490,  1372;
  1, 49,  735,  3430,   2401;
  1, 56, 1029,  6860,  12005;
  1, 63, 1372, 12005,  36015,  16807;
  1, 70, 1764, 19208,  84035, 100842;
  1, 77, 2205, 28812, 168070, 352947, 117649;
  1, 84, 2695, 41160, 302526, 941192, 823543;
		

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pages 70, 96.

Links

• Zagros Lalo, Left-justified triangle
• Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 7x)^n
• Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (7 + x)^n

Crossrefs

Row sums give A015442.

Cf. A013614, A027466.

Programs

GAP

Flat(List([0..13],n->List([0..Int(n/2)],k->7^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] + 7 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
Table[7^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten

Formula

T(n,k) = 7^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).