cp's OEIS Frontend

A204854 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^k) / (1 + x^k).

%I A204854 #13 Jun 18 2019 03:21:48
%S A204854 1,1,-1,1,-3,3,-3,3,-3,5,-5,5,-5,5,-5,5,-7,7,-7,7,-7,7,-7,7,-7,9,-9,9,
%T A204854 -9,9,-9,9,-9,9,-9,9,-11,11,-11,11,-11,11,-11,11,-11,11,-11,11,-11,13,
%U A204854 -13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-13,13,-15,15,-15,15,-15,15,-15,15
%N A204854 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^k) / (1 + x^k).
%H A204854 Seiichi Manyama, <a href="/A204854/b204854.txt">Table of n, a(n) for n = 0..10000</a>
%F A204854 G.f.: 1 - Sum_{n>=1} (-1)^n*(2*n-1)*x^(n^2)*(1+x^(2*n+1))/(1+x).
%F A204854 G.f.: (1 - Sum_{n>=1} (-1)^n*2*x^(n^2)) / (1+x).
%e A204854 G.f.: A(x) = 1 + x - x^2 + x^3 - 3*x^4 + 3*x^5 - 3*x^6 + 3*x^7 - 3*x^8 + 5*x^9 +...
%e A204854 where the g.f. is given by the series identities:
%e A204854 (0) A(x) = 1 + x*(1-x)/(1+x) + x^2*(1-x)*(1-x^2)/((1+x)*(1+x^2)) + x^3*(1-x)*(1-x^2)*(1-x^3)/((1+x)*(1+x^2)*(1+x^3)) + x^4*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...
%e A204854 (1) A(x) = 1 + x*(1+x^3)/(1+x) - 3*x^4*(1+x^5)/(1+x) + 5*x^9*(1+x^7)/(1+x) - 7*x^16*(1+x^9)/(1+x) + 9*x^25*(1+x^11)/(1+x) -+...
%e A204854 (2) A(x) = (1 + 2*x - 2*x^4 + 2*x^9 - 2*x^16 + 2*x^25 - 2*x^36 +...)/(1+x).
%o A204854 (PARI) {a(n)=polcoeff(1+sum(m=1,n,x^m*prod(k=1,m,(1-x^k)/(1+x^k+x*O(x^n)))),n)}
%Y A204854 Cf. A002448 (Jacobi theta_4), A207641.
%K A204854 sign
%O A204854 0,5
%A A204854 _Paul D. Hanna_, Jan 20 2012