A317052 Triangle read by rows: T(0,0) = 1; T(n,k) = 9 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, 9, 81, 1, 729, 18, 6561, 243, 1, 59049, 2916, 27, 531441, 32805, 486, 1, 4782969, 354294, 7290, 36, 43046721, 3720087, 98415, 810, 1, 387420489, 38263752, 1240029, 14580, 45, 3486784401, 387420489, 14880348, 229635, 1215, 1, 31381059609, 3874204890, 172186884, 3306744, 25515, 54

Offset

0,2

Comments

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013616 ((1+9*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038291 ((9+x)^n).

The coefficients in the expansion of 1/(1-9x-x^2) are given by the sequence generated by the row sums (see A099371).

If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 9.109772228646443655... (a metallic mean), when n approaches infinity; (see A176522: ((9+sqrt(85))/2)).

References

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 100.

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 + 9x)^n

Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (9 + x)^n

Example

Triangle begins:
1;
9;
81, 1;
729, 18;
6561, 243, 1;
59049, 2916, 27;
531441, 32805, 486, 1;
4782969, 354294, 7290, 36;
43046721, 3720087, 98415, 810, 1;
387420489, 38263752, 1240029, 14580, 45;
3486784401, 387420489, 14880348, 229635, 1215, 1;
31381059609, 3874204890, 172186884, 3306744, 25515, 54;
282429536481, 38354628411, 1937102445, 44641044, 459270, 1701, 1;
2541865828329, 376572715308, 21308126895, 573956280, 7440174, 40824, 63;

Mathematica

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 9 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten.

Prog

(PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 9*T(n-1, k)+T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

Crossrefs

Row sums give A099371.

Cf. A013616

Cf. A038291

Cf. A176522

Cf. A001019 (column 0), A053540 (column 1), A081139 (column 2), A173187 (column 3), A173000 (column 4).

Sequence in context: A127265 A055070 A143848 * A363420 A352386 A332702

Adjacent sequences: A317049 A317050 A317051 * A317053 A317054 A317055

Keyword

tabf,nonn,easy

Author

Zagros Lalo, Jul 20 2018

Status

approved