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A260148 Expansion of Sum_{n>=0} x^(n^2-n) / (1 + x^n)^n.

%I A260148 #19 Feb 20 2023 07:50:16
%S A260148 1,-1,2,-1,-1,-1,5,-1,-3,-4,6,-1,2,-1,8,-11,-11,-1,25,-1,2,-22,12,-1,
%T A260148 -3,-6,14,-37,22,-1,77,-1,-71,-56,18,-36,127,-1,20,-79,-69,-1,135,-1,
%U A260148 144,-232,24,-1,-179,-8,236,-137,261,-1,307,-331,-362,-172,30,-1,859,-1,32,-295,-599,-716,727,-1,647,-254,1247
%N A260148 Expansion of Sum_{n>=0} x^(n^2-n) / (1 + x^n)^n.
%H A260148 Paul D. Hanna, <a href="/A260148/b260148.txt">Table of n, a(n) for n = 0..2050</a>
%F A260148 G.f.: Sum_{n>=1} x^(-n) / (1 + x^(-n))^n.
%F A260148 G.f.: Sum_{n>=0} x^n * (x^n - 1)^n. - _Paul D. Hanna_, May 30 2018
%F A260148 a(p) = -1 for odd primes p.
%F A260148 a(n) = Sum_{d|n} (-1)^(d-n/d+1) * binomial(d,n/d-1) for n > 0. - _Seiichi Manyama_, Feb 20 2023
%e A260148 A(x) = 1 - x + 2*x^2 - x^3 - x^4 - x^5 + 5*x^6 - x^7 - 3*x^8 - 4*x^9 + 6*x^10 +...
%e A260148 where
%e A260148 A(x) = 1/(1+x) + x^2/(1+x^2)^2 + x^6/(1+x^3)^3 + x^12/(1+x^4)^4 + x^20/(1+x^5)^5 + ...
%e A260148 Also,
%e A260148 A(x) = 1 + x*(x-1) + x^2*(x^2-1)^2 + x^3*(x^3-1)^3 + x^4*(x^4-1)^4 + x^5*(x^5-1)^5 + ...
%o A260148 (PARI) a(n) = local(A=1); A = sum(k=1, sqrtint(n)+1, x^(k^2-k) / (1 + x^k +x*O(x^n) )^k ); polcoeff(A, n);
%o A260148 for(n=0, 70, print1(a(n), ", "))
%o A260148 (PARI) a(n) = local(A=0); A = sum(k=1,n+1, x^(-k)/(1 + x^(-k) +x*O(x^n) )^k ); polcoeff(A, n);
%o A260148 for(n=0, 70, print1(a(n), ", "))
%o A260148 (PARI) a(n) = if(n==0, 1, sumdiv(n, d, (-1)^(d-n/d+1)*binomial(d, n/d-1))); \\ _Seiichi Manyama_, Feb 20 2023
%Y A260148 Cf. A217668, A260147.
%K A260148 sign
%O A260148 0,3
%A A260148 _Paul D. Hanna_, Jul 17 2015