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 A356775 #9 Sep 23 2022 03:11:06
%S A356775 1,1,5,1,11,1,21,-8,36,1,22,1,85,-89,137,1,-23,1,302,-349,287,1,23,
%T A356775 -24,456,-944,1177,1,-903,1,2113,-2078,970,-559,709,1,1331,-4003,4293,
%U A356775 1,-3323,1,9153,-10694,2301,1,5869,-48,-4774,-11474,20294,1,-7334,-14783
%N A356775 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-2).
%C A356775 Related identities:
%C A356775 (I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).
%C A356775 (I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).
%C A356775 (I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).
%C A356775 (I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).
%C A356775 (I.5) 0 = Sum_{n=-oo..+oo} binomial(n+k-1, k) * x^(k*n) * (1 - x^n)^(n-1) for fixed positive integer k.
%C A356775 (I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).
%C A356775 (I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).
%C A356775 (I.8) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)/3! * x^(n*(n+2)) / (1 - x^(n+2))^(n+3).
%C A356775 (I.9) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)*(n+2)*(n+3)/4! * x^(n*(n+3)) / (1 - x^(n+3))^(n+4).
%C A356775 (I.10) 0 = Sum_{n=-oo..+oo} (-1)^n * binomial(n+k-1, k) * x^(n*(n+k-1)) / (1 - x^(n+k-1))^(n+k) for fixed positive integer k.
%C A356775 (I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)/6 * x^(2*n) * (1 - x^n)^(n-2).
%C A356775 (I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n-1)*n*(n+1)/6 * x^(n^2) / (1 - x^n)^(n+2).
%H A356775 Paul D. Hanna, <a href="/A356775/b356775.txt">Table of n, a(n) for n = 2..2050</a>
%F A356775 G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies:
%F A356775 (1) A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-2).
%F A356775 (2) A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(3*n) * (1 - x^n)^(n-2).
%F A356775 (3) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(2*n) * (1 - x^n)^(n-2).
%F A356775 (4) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n*(n-1)) / (1 - x^n)^(n+2).
%F A356775 (5) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n^2) / (1 - x^n)^(n+2).
%F A356775 (6) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)/6 * x^(n^2) / (1 - x^n)^(n+2).
%e A356775 G.f.: A(x) = x^2 + x^3 + 5*x^4 + x^5 + 11*x^6 + x^7 + 21*x^8 - 8*x^9 + 36*x^10 + x^11 + 22*x^12 + x^13 + 85*x^14 - 89*x^15 + 137*x^16 + ...
%e A356775 where
%e A356775 A(x) = ... + 3*x^(-6)*(1 - x^(-3))^(-5) + 1*x^(-4)*(1 - x^(-2))^(-4) + 0*x^(-2) + 0 + 1*x^2/(1-x) + 3*x^4 + 6*x^6*(1 - x^3) + 10*x^8*(1 - x^4)^2 + 15*x^10*(1 - x^5)^3 + ... + n*(n+1)/2 * x^(2*n)*(1 - x^n)^(n-2) + ...
%o A356775 (PARI) {a(n) = my(A = sum(m=-n-1,n+1, if(m==0,0, m*(m+1)/2 * x^(2*m) * (1 - x^m +x*O(x^n))^(m-2) )) );
%o A356775 polcoeff(A,n)}
%o A356775 for(n=2,100,print1(a(n),", "))
%Y A356775 Cf. A291937, A356774, A357156, A357157.
%K A356775 sign
%O A356775 2,3
%A A356775 _Paul D. Hanna_, Sep 22 2022