cp's OEIS Frontend

A136688 Triangle of polynomials F(x,n) = x*F(x,n-1) + 2*F(x,n-2).

%I A136688 #32 Jan 05 2025 19:51:38
%S A136688 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 A136688 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 A136688 280,0,112,0,18,0,1,0,192,0,560,0,448,0,144,0,20,0,1
%N A136688 Triangle of polynomials F(x,n) = x*F(x,n-1) + 2*F(x,n-2).
%C A136688 Riordan array (1/(1-2*x^2), x/(1-2*x^2)). - _Paul Barry_, Jun 18 2008
%C A136688 Antidiagonal sums are 1,0,3,0,9,... with g.f. 1/(1-3*x^2). - _Paul Barry_, Jun 18 2008
%H A136688 G. C. Greubel, <a href="/A136688/b136688.txt">Rows n = 1..100 of triangle, flattened</a> (terms 1..500 from Nathaniel Johnston)
%H A136688 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 A136688 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 A136688 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 A136688 From _Andrew Howroyd_, Feb 11 2023: (Start)
%F A136688 T(n,k) = 2^((n-k-1)/2)*binomial((n+k-1)/2, (n-k-1)/2) for k+1 == n (mod 2).
%F A136688 G.f.: x/(1 - y*x - 2*x^2). (End)
%e A136688 Triangle begins:
%e A136688    1;
%e A136688    0,   1;
%e A136688    2,   0,   1;
%e A136688    0,   4,   0,   1;
%e A136688    4,   0,   6,   0,   1;
%e A136688    0,  12,   0,   8,   0,   1;
%e A136688    8,   0,  24,   0,  10,   0,   1;
%e A136688    0,  32,   0,  40,   0,  12,   0,   1;
%e A136688   16,   0,  80,   0,  60,   0,  14,   0,   1;
%e A136688    0,  80,   0, 160,   0,  84,   0,  16,   0,   1;
%e A136688   32,   0, 240,   0, 280,   0, 112,   0,  18,   0,   1;
%e A136688   ...
%p A136688 A136688 := proc(n) option remember: if(n<=1)then return n: else return x*A136688(n-1)+2*A136688(n-2): fi: end:
%p A136688 seq(seq(coeff(A136688(n),x,m),m=0..n-1),n=1..10); # _Nathaniel Johnston_, Apr 27 2011
%t A136688 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 A136688 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 A136688 (Sage)
%o A136688 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 A136688 def A136688_list(prec):
%o A136688     P.<x> = PowerSeriesRing(ZZ, prec)
%o A136688     return P( f(n,x,2,1) ).list()
%o A136688 [A136688_list(n) for n in (1..10)] # _G. C. Greubel_, Dec 16 2019
%o A136688 (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 A136688 Row sums give A001045.
%Y A136688 Cf. A136689, A136705.
%K A136688 nonn,easy,tabl
%O A136688 1,4
%A A136688 _Roger L. Bagula_, Apr 06 2008