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 A356774 #20 Dec 25 2022 07:26:47
%S A356774 1,4,7,11,16,17,29,21,46,21,67,22,92,1,151,-23,154,22,191,-118,407,
%T A356774 -175,277,23,326,-363,946,-643,436,282,497,-1199,1948,-1019,701,-47,
%U A356774 704,-1519,3641,-3127,862,1759,947,-5301,7036,-2943,1129,-1187,1226,-2149,10252
%N A356774 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-2).
%C A356774 Related identities:
%C A356774 (I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).
%C A356774 (I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).
%C A356774 (I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).
%C A356774 (I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).
%C A356774 (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 A356774 (I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).
%C A356774 (I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).
%C A356774 (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 A356774 (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 A356774 (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 A356774 (I.11) 0 = Sum_{n=-oo..+oo} n*(n-1)/2 * x^n * (1 - x^n)^(n-2).
%C A356774 (I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^n)^(n+2).
%H A356774 Paul D. Hanna, <a href="/A356774/b356774.txt">Table of n, a(n) for n = 1..2050</a>
%F A356774 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F A356774 (1) A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-2).
%F A356774 (2) A(x) = Sum_{n=-oo..+oo} n * x^(2*n) * (1 - x^n)^(n-2).
%F A356774 (3.a) A(x) = Sum_{n=-oo..+oo} n*(n+1)/2 * x^n * (1 - x^n)^(n-2).
%F A356774 (3.b) A(x) = Sum_{n=-oo..+oo} n^2 * x^n * (1 - x^n)^(n-2).
%F A356774 (4) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n * x^(n^2) / (1 - x^n)^(n+2).
%F A356774 (5) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n * x^(n*(n+1)) / (1 - x^n)^(n+2).
%F A356774 (6.a) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)/2 * x^(n*(n+1)) / (1 - x^n)^(n+2).
%F A356774 (6.b) A(x) = Sum_{n=-oo..+oo} (-1)^n * n^2 * x^(n*(n+1)) / (1 - x^n)^(n+2).
%e A356774 G.f.: A(x) = x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 17*x^6 + 29*x^7 + 21*x^8 + 46*x^9 + 21*x^10 + 67*x^11 + 22*x^12 + 92*x^13 + x^14 + 151*x^15 + ...
%e A356774 where
%e A356774 A(x) = ... - 3*x^(-3)*(1 - x^(-3))^(-5) - 2*x^(-2)*(1 - x^(-2))^(-4) - x^(-1)*(1 - x^(-1))^(-3) + 0 + x/(1-x) + 2*x^2 + 3*x^3*(1 - x^3) + 4*x^4*(1 - x^4)^2 + 5*x^5*(1 - x^5)^3 + ... + n*x^n*(1 - x^n)^(n-2) + ...
%o A356774 (PARI) {a(n) = my(A = sum(m=-n-1,n+1, if(m==0,0, m * x^m * (1 - x^m +x*O(x^n))^(m-2) )) );
%o A356774 polcoeff(A,n)}
%o A356774 for(n=1,100,print1(a(n),", "))
%Y A356774 Cf. A291937, A356775, A357156, A357157.
%K A356774 sign
%O A356774 1,2
%A A356774 _Paul D. Hanna_, Sep 22 2022