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A267633 Expansion of (1 - 4t)/(1 - x + t x^2): a Fibonacci-type sequence of polynomials. 10
1, -4, 1, -4, 4, 1, -5, 4, 1, -6, 8, 1, -7, 13, -4, 1, -8, 19, -12, 1, -9, 26, -25, 4, 1, -10, 34, -44, 16, 1, -11, 43, -70, 41, 4, 1, -12, 53, -104, 85, -20, 1, -13, 64, -147, 155, -61, 4, 1, -14, 76, -200, 259, -146, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..54.

T. Copeland, Addendum to Elliptic Lie Triad

FORMULA

O.g.f. G(x,t) = (1 - 4t)/(1 - x + t x^2) = a /  [t (x - (1+sqrt(a))/(2t))(x - (1-sqrt(a))/(2t))] with a = 1-4t.

Recursion P(n,t) = -t P(n-2,t) + P(n-1,t) with P(-1,t)=0 and P(0,t) = 1-4t.

Convolution of the Fibonacci polynomials of signed A011973 Fb(n,-t) with coefficients of (1-4t), e.g., (1-4t)Fb(5,-t) = (1-4t)(1-3t+t^2) = 1-7t+13t^2-4t^3, so, for n>=1 and k<=(n-1), T(n,k) = (-1)^k [-4*binomial(n-(k-1),k-1) - binomial(n-k,k)] with the convention that 1/(-m)! = 0 for m>=1, i.e., let binomial(n,k) = nint[n!/((k+c)!(n-k+c)!)] for c sufficiently small in magnitude).

Third column is A034856, and the fourth, A000297. Embedded in the coefficients of the highest order term of the polynomials is A008586 (cf. also A008574).

With P(0,t)=0, the o.g.f. is H(x,t) = (1-4t) x(1-tx)/[1-x(1-tx)] = (1-4t) Linv(Cinv(tx)/t), where Linv(x) = x/(1-x) with inverse L(x) = x/(1+x) and Cinv(x) = x (1-x) is the inverse of the o.g.f. of the shifted Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2. Then Hinv(x,t) = C[t L(x/(1-4t))]/t = {1 - sqrt[1-4t(x/(1-4t))/[1+x/(1-4t)]]}/2t = {1-sqrt[1-4tx/(1-4t+x)]}/2t = 1/(1-4t) + (-1+t)/(1-4t)^2 x + (1-2t+2t^2)/(1-4t)^3 x^ + (-1+3t-6t^2+5t^3)/(1-4t)^4 + ..., where the numerators are the signed polynomials of A098474, reverse of A124644.

Row sums (t=1) are periodic -3,-3,0,3,3,0, repeat the six terms ... with o.g.f. -3 - 3x (1-x) / [1-x(1-x)]. Cf. A084103.

Unsigned row sums (t=-1) are shifted A022088 with o.g.f. 5 + 5x(1+x) / [x(1+x)].

EXAMPLE

Row polynomials:

P(0,t) = 1 - 4t

P(1,t) = 1 - 4t = [-t(0) + (1-4t)] = -t(0) + P(0,t)

P(2,t) = 1 - 5t + 4t^2 = [-t(1-4t) + (1-4t)] = -t P(0,t) + P(1,t)

P(3,t) = 1 - 6t + 8t^2 = [-t(1-4t) + (1-5t+4t^2)] = -t P(1,t) + P(2,t)

P(4,t) = 1 - 7t + 13t^2 - 4t^3 = [-t(1-5t+4t^2) + (1-6t+8t^2)]

P(5,t) = 1 - 8t + 19t^2 - 12t^3 = [-t(1-6t+8t^2) + (1-7t+13t^2)]

P(6,t) = 1 - 9t + 26t^2 - 25t^3 + 4t^4

P(7,t) = 1 - 10t + 34t^2 - 44t^3 + 16t^4

P(8,t) = 1 - 11t + 43t^2 - 70t^3 + 41t^4 - 4t^5

P(9,t) = 1 - 12t + 53t^2 - 104t^3 + 85t^4 - 20t^5

P(10,t) = 1 - 13t + 64t^2 - 147t^3 + 155t^4 - 61t^5 + 4t^6

P(11,t) = 1 - 14t + 76t^2 - 200t^3 + 259t^4 - 146t^5 + 24t^6

...

Apparently: The odd rows for n>1 are reversed rows of A140882 (mod signs), and the even rows for n>0, the 9th, 15th, 21st, 27th, etc. rows of A228785 (mod signs). The diagonals are reverse rows of A202241.

CROSSREFS

Cf.  A000108, A000297, A008574, A008586, A011973, A022088 A034856, A084103, A098474, A124644, A140882, A202241, A228785.

Sequence in context: A229705 A321593 A173259 * A021711 A334487 A327304

Adjacent sequences:  A267630 A267631 A267632 * A267634 A267635 A267636

KEYWORD

easy,sign,tabf

AUTHOR

Tom Copeland, Jan 18 2016

STATUS

approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)