A225200 Triangle (read by rows) of coefficients of the polynomials (in ascending order) of the denominators of the generalized sequence of 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, -1, 1, 1, -2, 2, -1, 1, 1, -4, 8, -10, 9, -6, 3, -1, 1, 1, -8, 32, -84, 162, -244, 298, -302, 258, -188, 118, -64, 30, -12, 4, -1, 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, 1

Offset

1,8

Comments

The degree of the polynomial in row n > 1 is 2^(n-2), hence the number of coefficients in row n > 1 is given by 2^(n-2) + 1 = A094373(n-1).

For n > 2 a new row always begins and ends with 1.

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..70.

Example

The triangle T(n,k), k = 0..2^(n-1), begins
   1;
  -1,  1;
   1, -1, 1;
   1, -2, 2,  -1, 1;
   1, -4, 8, -10, 9, -6, 3, -1, 1;

Maple

b:=n->m^(2^(n-2)); # n > 1
b(1):=m;
p:=proc(n) option remember; p(n-1)*a(n-1); end;
p(1):=1;
a:=proc(n) option remember; b(n)-p(n); end;
a(1):=1;
seq(op(PolynomialTools[CoefficientList](a(i), m, termorder=forward)), i=1..7);

Crossrefs

Cf. A100441, A225156, A225157, A225158, A225159, A225160, A225161, A225162, A225201.

Sequence in context: A340142 A242618 A180264 * A128706 A253586 A347621

Adjacent sequences: A225197 A225198 A225199 * A225201 A225202 A225203

Keyword

sign,tabf

Author

Martin Renner, May 01 2013

Status

approved