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A273894 Coefficients of iterations of polynomial x^2-x 1
1, -1, 1, 1, 0, -2, 1, -1, 1, 2, -5, 2, 4, -4, 1, 1, 0, -4, 2, 12, -14, -20, 48, -14, -50, 60, -10, -28, 24, -8, 1, -1, 1, 4, -10, -8, 54, -24, -180, 270, 270, -960, 150, 2064, -2040, -2352, 5871, -1566, -7236, 8880, 120, -9120, 7980, 120, -5340, 4212, -756 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Table T(n,k), n >=0, k=1..2^n, listed by rows.

Let p(0) = t, p(n) = p(n-1)^2 - p(n-1) for i >= 1.

T(n,k) is coefficient of t^k in p(n).

Rows sum to 0, except for row 0. - David A. Corneth, Jun 02 2016

LINKS

Robert Israel, Table of n, a(n) for n = 0..11212

FORMULA

T(n,k) = -T(n-1,k) + Sum_{j=1..k-1} T(n-1,j) T(n-1,k-j).

Column k is of the form

T(n,k) = b_k(n) + (-1)^n*c_k(n)

where b_k and c_k seem to be polynomials of degree floor(k/2) - 1 and floor((k-1)/2) respectively (except b_1 = 0).

Leading coefficient of b_k(n) + (-1)^n*c_k(n) seems to be

  -(-2)^(k/2-2) - binomial(-3/2,k/2-1)*2^(k/2-2)*(-1)^n if k is even,

  2^((k-1)/2)*binomial(-1/2,(k-1)/2)*(-1)^n if k is odd.

T(n,1) = (-1)^n = A033999(n).

T(n,2) = 1/2 + (-1)^n/2 = A000035(n)

T(n,3) = -1/2 + (-n + 1/2)*(-1)^n = -A137501(n).

T(n,4) = -n + 5/4 + (3*n/2 - 5/4)*(-1)^n)

  = A001477(n/2) if n is even, -5*A001477((n-1)/2) if n is odd.

T(n,5) = 2*n - 11/4 + (3*n^2/2 - 5*n + 11/4)*(-1)^n

  = 12*A161680(n/2) if n is even, -2*A270710((n-3)/2) if n >= 3 is odd.

T(n, 2^n) = 1 = A000012(n). - David A. Corneth, Jun 02 2016

EXAMPLE

Table starts

1

-1, 1

1, 0, -2, 1

-1, 1, 2, -5, 2, 4, -4, 1

1, 0, -4, 2, 12, -14, -20, 48, -14, -50, 60, -10, -28, 24, -8, 1

MAPLE

P[0]:= t:

for n from 1 to 8 do

  P[n]:= expand(P[n-1]^2 - P[n-1])

od:

seq(seq(coeff(P[n], t, j), j=1..2^n), n=0..8);

MATHEMATICA

CoefficientList[NestList[Expand[#^2-#]&, x, 5]/x, x] // Flatten (* Jean-Fran├žois Alcover, Apr 29 2019 *)

CROSSREFS

Cf. A000035, A001477, A033999, A137501, A161680, A270710.

Sequence in context: A287641 A265312 A241531 * A308035 A221131 A126886

Adjacent sequences:  A273891 A273892 A273893 * A273895 A273896 A273897

KEYWORD

sign,tabf

AUTHOR

Robert Israel, Jun 02 2016

STATUS

approved

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)