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
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
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Jul 20 2018
STATUS
approved