A273894 - OEIS

%I #29 Jun 03 2024 18:43:46

%S 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,

%T -10,-28,24,-8,1,-1,1,4,-10,-8,54,-24,-180,270,270,-960,150,2064,

%U -2040,-2352,5871,-1566,-7236,8880,120,-9120,7980,120,-5340,4212,-756

%N 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.

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

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

%C Rows sum to 0, except for row 0. - _David A. Corneth_, Jun 02 2016

%H Robert Israel, <a href="/A273894/b273894.txt">Table of n, a(n) for n = 0..11212</a>

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

%F Column k is of the form

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

%F 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).

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

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

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

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

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

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

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

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

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

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

%F T(n, 2^n) = 1 = A000012(n). - _David A. Corneth_, Jun 02 2016

%e Table starts

%e 1

%e -1, 1

%e 1, 0, -2, 1

%e -1, 1, 2, -5, 2, 4, -4, 1

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

%p P[0]:= t:

%p for n from 1 to 8 do

%p   P[n]:= expand(P[n-1]^2 - P[n-1])

%p od:

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

%t CoefficientList[NestList[Expand[#^2-#]&, x, 5]/x, x] // Flatten (* _Jean-François Alcover_, Apr 29 2019 *)

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

%K sign,tabf

%O 0,6

%A _Robert Israel_, Jun 02 2016