A136688 Triangle of polynomials F(x,n) = x*F(x,n-1) + 2*F(x,n-2).
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, 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, 280, 0, 112, 0, 18, 0, 1, 0, 192, 0, 560, 0, 448, 0, 144, 0, 20, 0, 1
Offset: 1
Examples
Triangle begins: 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, 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, 280, 0, 112, 0, 18, 0, 1; ...
Links
- G. C. Greubel, Rows n = 1..100 of triangle, flattened (terms 1..500 from Nathaniel Johnston)
- J. Cigler, q-Fibonacci polynomials, Fibonacci Quarterly 41 (2003) 31-40.
Programs
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Maple
A136688 := proc(n) option remember: if(n<=1)then return n: else return x*A136688(n-1)+2*A136688(n-2): fi: end: seq(seq(coeff(A136688(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,12}]//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,2,1], x], {n, 1, 10}]//Flatten (* G. C. Greubel, Dec 16 2019 *) -
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
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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 A136688_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P( f(n,x,2,1) ).list() [A136688_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=2.
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
From Andrew Howroyd, Feb 11 2023: (Start)
T(n,k) = 2^((n-k-1)/2)*binomial((n+k-1)/2, (n-k-1)/2) for k+1 == n (mod 2).
G.f.: x/(1 - y*x - 2*x^2). (End)
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