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

Author

Zagros Lalo, Jul 20 2018

Keywords

• tabf
• nonn
• easy

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/(1-9*x-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)).

Examples

			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;
		

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

• 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

Crossrefs

Row sums give A099371.

Cf. A013616, A038291, A176522.

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

Programs

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

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