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%I #30 Feb 17 2023 22:33:50
%S 1,0,1,2,0,1,0,4,0,1,4,0,6,0,1,0,12,0,8,0,1,8,0,24,0,10,0,1,0,32,0,40,
%T 0,12,0,1,16,0,80,0,60,0,14,0,1,0,80,0,160,0,84,0,16,0,1,32,0,240,0,
%U 280,0,112,0,18,0,1,0,192,0,560,0,448,0,144,0,20,0,1
%N Triangle of polynomials F(x,n) = x*F(x,n-1) + 2*F(x,n-2).
%C Riordan array (1/(1-2*x^2), x/(1-2*x^2)). - _Paul Barry_, Jun 18 2008
%C Antidiagonal sums are 1,0,3,0,9,... with g.f. 1/(1-3*x^2). - _Paul Barry_, Jun 18 2008
%H G. C. Greubel, <a href="/A136688/b136688.txt">Rows n = 1..100 of triangle, flattened</a> (terms 1..500 from Nathaniel Johnston)
%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=2.
%F F(x,n) = Sum_{j=0..floor((n-1)/2)} binomial(n-j-1, j)*x^(n-2*j-1)*2^j, for n >= 1. See the Mma program by _G. C. Greubel_. - _Wolfdieter Lang_, Feb 10 2023
%F From _Andrew Howroyd_, Feb 11 2023: (Start)
%F T(n,k) = 2^((n-k-1)/2)*binomial((n+k-1)/2, (n-k-1)/2) for k+1 == n (mod 2).
%F G.f.: x/(1 - y*x - 2*x^2). (End)
%e Triangle begins:
%e 1;
%e 0, 1;
%e 2, 0, 1;
%e 0, 4, 0, 1;
%e 4, 0, 6, 0, 1;
%e 0, 12, 0, 8, 0, 1;
%e 8, 0, 24, 0, 10, 0, 1;
%e 0, 32, 0, 40, 0, 12, 0, 1;
%e 16, 0, 80, 0, 60, 0, 14, 0, 1;
%e 0, 80, 0, 160, 0, 84, 0, 16, 0, 1;
%e 32, 0, 240, 0, 280, 0, 112, 0, 18, 0, 1;
%e ...
%p A136688 := proc(n) option remember: if(n<=1)then return n: else return x*A136688(n-1)+2*A136688(n-2): fi: end:
%p seq(seq(coeff(A136688(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[CoefficientList[F[x, n], x], {n,12}]//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,2,1], 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 A136688_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( f(n,x,2,1) ).list()
%o [A136688_list(n) for n in (1..10)] # _G. C. Greubel_, Dec 16 2019
%o (PARI) T(n,k)=if((n-k)%2==0, 0, 2^((n-k-1)/2)*binomial((n+k-1)/2, (n-k-1)/2)) \\ _Andrew Howroyd_, Feb 11 2023
%Y Row sums give A001045.
%Y Cf. A136689, A136705.
%K nonn,easy,tabl
%O 1,4
%A _Roger L. Bagula_, Apr 06 2008
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