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%I #18 Dec 16 2019 22:10:22
%S 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 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 1215,0,945,0,252,0,27,0,1,0,1458,0,2835,0,1512,0
%N Triangular sequence of q-Fibonacci polynomials for s=3: F(x,n) = x*F(x,n-1) + s*F(x,n-2).
%C Row sums: 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ... = A006130(n-1).
%H Nathaniel Johnston, <a href="/A136689/b136689.txt">Rows n=1..36 of triangle, flattened</a>
%H J. Cigler, <a href="http://www.fq.math.ca/Scanned/41-1/cigler.pdf">q-Fibonacci polynomials</a>, Fibonacci Quarterly 41 (2003) 31-40.
%F 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 Triangle begins:
%e 1;
%e 0, 1;
%e 3, 0, 1;
%e 0, 6, 0, 1;
%e 9, 0, 9, 0, 1;
%e 0, 27, 0, 12, 0, 1;
%e 27, 0, 54, 0, 15, 0, 1;
%e 0, 108, 0, 90, 0, 18, 0, 1;
%e 81, 0, 270, 0, 135, 0, 21, 0, 1;
%e 0, 405, 0, 540, 0, 189, 0, 24, 0, 1;
%e 243, 0, 1215, 0, 945, 0, 252, 0, 27, 0, 1;
%e ...
%p A136689 := proc(n) option remember: if(n<=1)then return n: else return x*procname(n-1)+3*procname(n-2): fi: end:
%p seq(seq(coeff(A136689(n), x, m), m=0..n-1), n=1..10); # Nathaniel Johnston, Apr 27 2011
%t s=2; F[x_, n_]:= F[x, n]= If[n<2, n, x*F[x, n-1] + s*F[x, n-2]]; Table[
%t CoefficientList[F[x, n], x], {n,10}]//Flatten
%t 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 (Sage)
%o 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 def A136689_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( f(n,x,3,1) ).list()
%o [A136689_list(n) for n in (1..10)] # _G. C. Greubel_, Dec 16 2019
%Y Cf. A136688, A136705.
%K nonn,easy,tabl
%O 1,4
%A _Roger L. Bagula_, Apr 06 2008
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