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
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.
Programs
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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
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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.
Comments