A225201 Triangle (read by rows) of coefficients of the polynomials (in ascending order) of the denominators of the generalized sequence s(n) of the sum resp. product of generalized fractions f(n) defined recursively by f(1) = m/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

1, -1, 1, -1, 2, -2, 1, -1, 4, -8, 10, -9, 6, -3, 1, -1, 8, -32, 84, -162, 244, -298, 302, -258, 188, -118, 64, -30, 12, -4, 1, -1, 16, -128, 680, -2692, 8456, -21924, 48204, -91656, 152952, -226580, 300664, -359992, 391232, -387820, 352074, -293685, 225696, -160120, 105024, -63750, 35832, -18654, 8994, -4014, 1656, -630, 220, -70, 20, -5, 1

Offset

1,5

Comments

The degree of the polynomial in row n > 1 is 2^(n-1) - 1, hence the number of coefficients in row n >= 1 is given by 2^(n-1) = A000079(n-1).

For n > 1 a new row begins always with -1 and ends always with 1.

The coefficients in row n are the first k negative coefficients in row n+1 in A225200.

The sum and product of the generalized sequence of fractions given by m^(2^(n-2)) divided by the polynomial p(n) are equal, i.e.,

m + m/(m-1) = m * m/(m-1) = m^2/(m-1);

m + m/(m-1) + m^2/(m^2-m+1) = m * (m/(m-1)) * m^2/(m^2-m+1) = m^4/(m^3-2*m^2+2*m-1).

Links

Table of n, a(n) for n=1..63.

Example

The triangle T(n,k), k = 0..2^(n-1)-1, begins
   1;
  -1, 1;
  -1, 2,  -2,  1;
  -1, 4,  -8, 10,   -9,   6,   -3,   1;
  -1, 8, -32, 84, -162, 244, -298, 302, -258, 188, -118, 64, -30, 12, -4, 1;

Maple

b:=proc(n) option remember; b(n-1)-b(n-1)^2; end;
b(1):=1/m;
a:=n->m^(2^(i-1))*normal(b(i));
seq(op(PolynomialTools[CoefficientList](a(i), m, termorder=forward)), i=1..6);

Crossrefs

Cf. A076628, A225163 to A225169, A225200.

Sequence in context: A297347 A342623 A121697 * A124976 A176663 A113021

Adjacent sequences: A225198 A225199 A225200 * A225202 A225203 A225204

Keyword

sign,tabf

Author

Martin Renner, May 01 2013

Status

approved