A225200 -id:A225200 - cp's OEIS frontend

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

Martin Renner, May 01 2013

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).

			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;

Cf. A076628, A225163 to A225169, A225200.

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);

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