A273894 Table T(n,k), n >= 0, k = 1..2^n, read by rows, giving coefficients of iterations of polynomial x^2-x: see Comments for precise definition.

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

Offset: 0

Author

Robert Israel, Jun 02 2016

Keywords

• sign
• tabf

Comments

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

Examples

			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;
  ...
		

Links

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

Crossrefs

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

Programs

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-François Alcover, Apr 29 2019 *)

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

= n/2 if n is even, -5*(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