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.

%I A225201 #10 Jun 25 2023 08:10:02
%S A225201 1,-1,1,-1,2,-2,1,-1,4,-8,10,-9,6,-3,1,-1,8,-32,84,-162,244,-298,302,
%T A225201 -258,188,-118,64,-30,12,-4,1,-1,16,-128,680,-2692,8456,-21924,48204,
%U A225201 -91656,152952,-226580,300664,-359992,391232,-387820,352074,-293685,225696,-160120,105024,-63750,35832,-18654,8994,-4014,1656,-630,220,-70,20,-5,1
%N 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.
%C A225201 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).
%C A225201 For n > 1 a new row begins always with -1 and ends always with 1.
%C A225201 The coefficients in row n are the first k negative coefficients in row n+1 in A225200.
%C A225201 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.,
%C A225201 m + m/(m-1) = m * m/(m-1) = m^2/(m-1);
%C A225201 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).
%e A225201 The triangle T(n,k), k = 0..2^(n-1)-1, begins
%e A225201    1;
%e A225201   -1, 1;
%e A225201   -1, 2,  -2,  1;
%e A225201   -1, 4,  -8, 10,   -9,   6,   -3,   1;
%e A225201   -1, 8, -32, 84, -162, 244, -298, 302, -258, 188, -118, 64, -30, 12, -4, 1;
%p A225201 b:=proc(n) option remember; b(n-1)-b(n-1)^2; end;
%p A225201 b(1):=1/m;
%p A225201 a:=n->m^(2^(i-1))*normal(b(i));
%p A225201 seq(op(PolynomialTools[CoefficientList](a(i),m,termorder=forward)),i=1..6);
%Y A225201 Cf. A076628, A225163 to A225169, A225200.
%K A225201 sign,tabf
%O A225201 1,5
%A A225201 _Martin Renner_, May 01 2013