A317014 - OEIS

A317014

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

1, 7, 49, 1, 343, 14, 2401, 147, 1, 16807, 1372, 21, 117649, 12005, 294, 1, 823543, 100842, 3430, 28, 5764801, 823543, 36015, 490, 1, 40353607, 6588344, 352947, 6860, 35, 282475249, 51883209, 3294172, 84035, 735, 1, 1977326743, 403536070, 29647548, 941192, 12005, 42

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OFFSET

0,2

COMMENTS

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013614 ((1+7*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A027466 ((7+x)^n).

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

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 7.14005494464025913554... ((7+sqrt(53))/2), a metallic mean (see A176439), 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, pp. 70, 96.

LINKS

Table of n, a(n) for n=0..41.

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

EXAMPLE

Triangle begins:

49, 1;

343, 14;

2401, 147, 1;