This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A261605 #19 Mar 03 2025 13:31:55
%S A261605 1,2,-1,4,-1,6,-6,8,2,12,-15,12,8,14,-28,32,22,18,-55,20,34,72,-66,24,
%T A261605 44,28,-91,140,62,30,-205,32,209,244,-153,72,-98,38,-190,392,443,42,
%U A261605 -518,44,-1,788,-276,48,506,52,-451,852,-196,54,-1086,728,1636,1180,-435,60,-1691
%N A261605 G.f.: Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n)^n.
%H A261605 Paul D. Hanna, <a href="/A261605/b261605.txt">Table of n, a(n) for n = 0..1000</a>
%F A261605 G.f.: Sum_{n=-oo..+oo} (x^n - 1)^n.
%F A261605 G.f.: 1/2 + Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n)^(n+1).
%F A261605 G.f.: 1/2 + Sum_{n=-oo..+oo} x^n * (1 + x^n)^(n-1).
%F A261605 From _Peter Bala_, Mar 03 2025: (Start)
%F A261605 For n >= 1, a(n) = Sum_{d divides n} (-1)^(n/d+d)*binomial(d, n/d) + binomial(n/d-1, d-1).
%F A261605 For odd prime p, a(p) = p + 1, a(2*p) = - p*(p + 1)/2, a(p^2) = p^2 + 3. (End)
%e A261605 G.f.: A(x) = 1 + 2*x - x^2 + 4*x^3 - x^4 + 6*x^5 - 6*x^6 + 8*x^7 + 2*x^8 + 12*x^9 - 15*x^10 + 12*x^11 + 8*x^12 + 14*x^13 - 28*x^14 + 32*x^15 + 22*x^16 +...
%e A261605 where A(x) = 1 + N(x) + P(x) such that
%e A261605 N(x) = (x-1) + (x^2-1)^2 + (x^3-1)^3 + (x^4-1)^4 + (x^5-1)^5 + (x^6-1)^6 +...
%e A261605 P(x) = x/(1-x) + x^4/(1-x^2)^2 + x^9/(1-x^3)^3 + x^16/(1-x^4)^4 + x^25/(1-x^5)^5 +...
%e A261605 explicitly,
%e A261605 N(x) = x - 2*x^2 + 3*x^3 - 3*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 2*x^8 + 10*x^9 - 20*x^10 + 11*x^11 - x^12 + 13*x^13 - 35*x^14 + 25*x^15 + 13*x^16 +...
%e A261605 P(x) = x + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + x^7 + 4*x^8 + 2*x^9 + 5*x^10 + x^11 + 9*x^12 + x^13 + 7*x^14 + 7*x^15 + 9*x^16 +...+ A143862(n)*x^n +...
%p A261605 with(numtheory):
%p A261605 seq(add( (-1)^(n/d+d)*binomial(d, n/d) + binomial(n/d-1, d-1), d in divisors(n)), n = 1..60); # _Peter Bala_, Mar 03 2025
%o A261605 (PARI) {a(n) = polcoeff(sum(m=-n-2,n+2,x^(m^2)/(1-x^m +x*O(x^n))^m), n)}
%o A261605 for(n=0,60,print1(a(n),", "))
%o A261605 (PARI) {a(n) = polcoeff(sum(m=-n-2,n+2,(x^m-1 +x*O(x^n))^m), n)}
%o A261605 for(n=0,60,print1(a(n),", "))
%o A261605 (PARI) {a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^(m^2)/(1+x^m +x*O(x^n))^(m+1)), n)}
%o A261605 for(n=0,60,print1(a(n),", "))
%o A261605 (PARI) {a(n) = polcoeff(1/2 + sum(m=-n-2,n+2,x^m*(1+x^m +x*O(x^n))^(m-1)), n)}
%o A261605 for(n=0,60,print1(a(n),", "))
%Y A261605 Cf. A143862, A260147, A261608.
%K A261605 sign,easy
%O A261605 0,2
%A A261605 _Paul D. Hanna_, Aug 25 2015