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A136688 Triangular sequence of q-Fibonacci polynomials for s=2: F(x,n) = x*F(x,n-1) + s*F(x,n-2). 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums are: 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, ... = A001045(n).

Riordan array (1/(1-2*x^2), x/(1-2*x^2)). - Paul Barry, Jun 18 2008

Diagonal sums are 1,0,3,0,9,... with g.f. 1/(1-3*x^2). - Paul Barry, Jun 18 2008

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.

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.

EXAMPLE

Triangle begins as:

   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;

...

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

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 *)

PROG

(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.<x> = 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

CROSSREFS

Cf. A136689, A136705.

Sequence in context: A073430 A053389 A202328 * A131321 A111959 A110109

Adjacent sequences:  A136685 A136686 A136687 * A136689 A136690 A136691

KEYWORD

nonn,easy,tabl

AUTHOR

Roger L. Bagula, Apr 06 2008

STATUS

approved

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Last modified March 29 21:32 EDT 2020. Contains 333117 sequences. (Running on oeis4.)