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

%I A136689 #20 Jan 05 2025 19:51:38
%S A136689 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,
%T A136689 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,
%U A136689 1215,0,945,0,252,0,27,0,1,0,1458,0,2835,0,1512,0
%N A136689 Triangular sequence of q-Fibonacci polynomials for s=3: F(x,n) = x*F(x,n-1) + s*F(x,n-2).
%C A136689 Row sums: 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ... = A006130(n-1).
%H A136689 Nathaniel Johnston, <a href="/A136689/b136689.txt">Rows n=1..36 of triangle, flattened</a>
%H A136689 J. Cigler, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/41-1/cigler.pdf">q-Fibonacci polynomials</a>, Fibonacci Quarterly 41 (2003) 31-40.
%F A136689 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.
%e A136689 Triangle begins:
%e A136689     1;
%e A136689     0,   1;
%e A136689     3,   0,    1;
%e A136689     0,   6,    0,   1;
%e A136689     9,   0,    9,   0,   1;
%e A136689     0,  27,    0,  12,   0,   1;
%e A136689    27,   0,   54,   0,  15,   0,   1;
%e A136689     0, 108,    0,  90,   0,  18,   0,  1;
%e A136689    81,   0,  270,   0, 135,   0,  21,  0,  1;
%e A136689     0, 405,    0, 540,   0, 189,   0, 24,  0, 1;
%e A136689   243,   0, 1215,   0, 945,   0, 252,  0, 27, 0, 1;
%e A136689   ...
%p A136689 A136689 := proc(n) option remember: if(n<=1)then return n: else return x*procname(n-1)+3*procname(n-2): fi: end:
%p A136689 seq(seq(coeff(A136689(n), x, m), m=0..n-1), n=1..10); # Nathaniel Johnston, Apr 27 2011
%t A136689 s=2; F[x_, n_]:= F[x, n]= If[n<2, n, x*F[x, n-1] + s*F[x, n-2]]; Table[
%t A136689 CoefficientList[F[x, n], x], {n,10}]//Flatten
%t A136689 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 *)
%o A136689 (Sage)
%o A136689 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)))
%o A136689 def A136689_list(prec):
%o A136689     P.<x> = PowerSeriesRing(ZZ, prec)
%o A136689     return P( f(n,x,3,1) ).list()
%o A136689 [A136689_list(n) for n in (1..10)] # _G. C. Greubel_, Dec 16 2019
%Y A136689 Cf. A136688, A136705.
%K A136689 nonn,easy,tabl
%O A136689 1,4
%A A136689 _Roger L. Bagula_, Apr 06 2008