OFFSET
0,4
COMMENTS
The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013616 ((1+9*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A038291 ((9+x)^n).
The coefficients in the expansion of 1/(1-x-9x^2) are given by the sequence generated by the row sums.
The row sums are Generalized Fibonacci numbers (see A015445).
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.5413812651491... ((1+sqrt(37))/2), 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, 100
LINKS
FORMULA
T(n,k) = 9^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).
EXAMPLE
Triangle begins:
1;
1;
1, 9;
1, 18;
1, 27, 81;
1, 36, 243;
1, 45, 486, 729;
1, 54, 810, 2916;
1, 63, 1215, 7290, 6561;
1, 72, 1701, 14580, 32805;
1, 81, 2268, 25515, 98415, 59049;
1, 90, 2916, 40824, 229635, 354294;
1, 99, 3645, 61236, 459270, 1240029, 531441;
1, 108, 4455, 87480, 826686, 3306744, 3720087;
1, 117, 5346, 120285, 1377810, 7440174, 14880348, 4782969;
1, 126, 6318, 160380, 2165130, 14880348, 44641044, 38263752;
1, 135, 7371, 208494, 3247695, 27280638, 111602610, 172186884, 43046721;
MATHEMATICA
t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 9 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten.
Table[9^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->9^k*Binomial(n-k, k)))); # Muniru A Asiru, Jul 20 2018
(PARI) T(n, k) = 9^k*binomial(n-k, k);
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018
(Magma) /* As triangle */ [[9^k*Binomial(n-k, k): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 05 2018
CROSSREFS
KEYWORD
tabf,nonn,easy
AUTHOR
Zagros Lalo, Jul 20 2018
STATUS
approved