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

0,4

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-7x^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).

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

Table of n, a(n) for n=0..55.

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

Formula

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

Example

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;
  1, 91, 3234, 56595, 504210, 2117682, 3294172,  823543;
  1, 98, 3822, 75460, 792330, 4235364, 9882516, 6588344;

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, 15}, {k, 0, Floor[n/2]}] // Flatten.
Table[7^k Binomial[n - k, k], {n, 0, 15}, {k, 0, Floor[n/2]}].

Prog

(GAP) Flat(List([0..13], n->List([0..Int(n/2)], k->7^k*Binomial(n-k, k)))); # Muniru A Asiru, Jul 19 2018

Crossrefs

Row sums give A015442.

Cf. A013614

Cf. A027466

Sequence in context: A225017 A268919 A040055 * A348983 A013614 A179530

Adjacent sequences: A317013 A317014 A317015 * A317017 A317018 A317019

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 19 2018

Status

approved