A136689 Triangular sequence of q-Fibonacci polynomials for s=3: F(x,n) = x*F(x,n-1) + s*F(x,n-2).

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, 0, 1458, 0, 2835, 0, 1512, 0

Offset: 1

Author

Roger L. Bagula, Apr 06 2008

Keywords

• nonn
• easy
• tabl

Comments

Row sums: 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ... = A006130(n-1).

Examples

			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;
  ...
		

Links

• Nathaniel Johnston, Rows n=1..36 of triangle, flattened
• J. Cigler, q-Fibonacci polynomials, Fibonacci Quarterly 41 (2003) 31-40.

Crossrefs

Cf. A136688, A136705.

Programs

Maple

A136689 := proc(n) option remember: if(n<=1)then return n: else return x*procname(n-1)+3*procname(n-2): fi: end:
seq(seq(coeff(A136689(n), x, m), m=0..n-1), n=1..10); # Nathaniel Johnston, Apr 27 2011

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 *)

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. = 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

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.