OFFSET
1,4
COMMENTS
Row sums: 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ... = A006130(n-1).
LINKS
Nathaniel Johnston, Rows n=1..36 of triangle, flattened
J. Cigler, q-Fibonacci polynomials, Fibonacci Quarterly 41 (2003) 31-40.
FORMULA
F(x,n) = x*F(x,n-1) + s*F(x,n-2), where F(x,0)=0, F(x,1)=1 and s=3.
EXAMPLE
Triangle begins:
1;
0, 1;
3, 0, 1;
0, 6, 0, 1;
9, 0, 9, 0, 1;
0, 27, 0, 12, 0, 1;
27, 0, 54, 0, 15, 0, 1;
0, 108, 0, 90, 0, 18, 0, 1;
81, 0, 270, 0, 135, 0, 21, 0, 1;
0, 405, 0, 540, 0, 189, 0, 24, 0, 1;
243, 0, 1215, 0, 945, 0, 252, 0, 27, 0, 1;
...
MAPLE
MATHEMATICA
s=2; F[x_, n_]:= F[x, n]= If[n<2, n, x*F[x, n-1] + s*F[x, n-2]]; Table[
CoefficientList[F[x, n], x], {n, 10}]//Flatten
F[n_, x_, s_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^Binomial[j+1, 2]*x^(n-2*j-1) *s^j, {j, 0, Floor[(n-1)/2]}]; Table[CoefficientList[F[n, x, 3, 1], x], {n, 1, 10}]//Flatten (* G. C. Greubel, Dec 16 2019 *)
PROG
(Sage)
def f(n, x, s, q): return sum( q_binomial(n-j-1, j, q)*q^binomial(j+1, 2)*x^(n-2*j-1)*s^j for j in (0..floor((n-1)/2)))
def A136689_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(n, x, 3, 1) ).list()
[A136689_list(n) for n in (1..10)] # G. C. Greubel, Dec 16 2019
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Apr 06 2008
STATUS
approved