%I #9 Sep 23 2022 03:11:13

%S 1,1,1,6,1,1,16,1,1,22,1,1,71,-63,1,127,1,-158,211,1,1,-117,176,1,496,

%T -923,1,1277,1,-1727,1002,1,1681,-2021,1,1,1821,-1027,1,912,1,-7721,

%U 11146,1,1,-12571,736,15401,4846,-17016,1,-6389,27457,-20956,7316,1,1,-6486,1,1,22177

%N Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(3*n) * (1 - x^n)^(n-2).

%C Related identities:

%C (I.1) 0 = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-1).

%C (I.2) 0 = Sum_{n=-oo..+oo} n*(n+1)/2 * x^(2*n) * (1 - x^n)^(n-1).

%C (I.3) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/3! * x^(3*n) * (1 - x^n)^(n-1).

%C (I.4) 0 = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/4! * x^(4*n) * (1 - x^n)^(n-1).

%C (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 (I.6) 0 = Sum_{n=-oo..+oo} (-1)^n * n * x^(n^2) / (1 - x^n)^(n+1).

%C (I.7) 0 = Sum_{n=-oo..+oo} (-1)^n * n*(n+1)/2 * x^(n*(n+1)) / (1 - x^(n+1))^(n+2).

%C (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 (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 (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 (I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)*(n+2)/24 * x^(3*n) * (1 - x^n)^(n-2).

%C (I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n+1)*n*(n-1)*(n-2)/24 * x^(n*(n-1)) / (1 - x^n)^(n+2).

%H Paul D. Hanna, <a href="/A357156/b357156.txt">Table of n, a(n) for n = 3..2050</a>

%F G.f. A(x) = Sum_{n>=3} a(n)*x^n satisfies:

%F (1) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(3*n) * (1 - x^n)^(n-2).

%F (2) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)/6 * x^(4*n) * (1 - x^n)^(n-2).

%F (3) A(x) = Sum_{n=-oo..+oo} n*(n+1)*(n+2)*(n+3)/24 * x^(3*n) * (1 - x^n)^(n-2).

%F (4) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)/6 * x^(n*(n-2)) / (1 - x^n)^(n+2).

%F (5) A(x) = Sum_{n=-oo..+oo} -(-1)^n * n*(n-1)*(n-2)/6 * x^(n*(n-1)) / (1 - x^n)^(n+2).

%F (6) A(x) = Sum_{n=-oo..+oo} (-1)^n * n*(n-1)*(n-2)*(n-3)/24 * x^(n*(n-1)) / (1 - x^n)^(n+2).

%e G.f.: A(x) = x^3 + x^4 + x^5 + 6*x^6 + x^7 + x^8 + 16*x^9 + x^10 + x^11 + 22*x^12 + x^13 + x^14 + 71*x^15 - 63*x^16 + x^17 + 127*x^18 + ...

%e where

%e A(x) = ... - 4*x^(-12)*(1 - x^(-4))^(-6) - 1*x^(-9)*(1 - x^(-3))^(-5) + 0*x^(-6) + 0*x^(-3) + 0 + 1*x^3/(1-x) + 4*x^6 + 10*x^9*(1 - x^3) + 20*x^12*(1 - x^4)^2 + 35*x^15*(1 - x^5)^3 + ... + n*(n+1)*(n+2)/6 * x^(3*n)*(1 - x^n)^(n-2) + ...

%o (PARI) {a(n) = my(A = sum(m=-n-1,n+1, if(m==0,0, m*(m+1)*(m+2)/6 * x^(3*m) * (1 - x^m +x*O(x^n))^(m-2) )) );

%o polcoeff(A,n)}

%o for(n=3,100,print1(a(n),", "))

%Y Cf. A291937, A356774, A356775, A357157.

%K sign

%O 3,4

%A _Paul D. Hanna_, Sep 22 2022