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).

1, 1, 5, 1, 11, 1, 21, -8, 36, 1, 22, 1, 85, -89, 137, 1, -23, 1, 302, -349, 287, 1, 23, -24, 456, -944, 1177, 1, -903, 1, 2113, -2078, 970, -559, 709, 1, 1331, -4003, 4293, 1, -3323, 1, 9153, -10694, 2301, 1, 5869, -48, -4774, -11474, 20294, 1, -7334, -14783

Offset

2,3

Comments

Related identities:

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

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

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

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

(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.

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

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

(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).

(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).

(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.

(I.11) 0 = Sum_{n=-oo..+oo} (n-1)*n*(n+1)/6 * x^(2*n) * (1 - x^n)^(n-2).

(I.12) 0 = Sum_{n=-oo..+oo} (-1)^n * (n-1)*n*(n+1)/6 * x^(n^2) / (1 - x^n)^(n+2).

Links

Paul D. Hanna, Table of n, a(n) for n = 2..2050

Formula

G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies:

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

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

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

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

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

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

Example

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 + ...
where
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) + ...

Prog

(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) )) );
polcoeff(A, n)}
for(n=2, 100, print1(a(n), ", "))

Crossrefs

Cf. A291937, A356774, A357156, A357157.

Sequence in context: A189745 A062967 A245211 * A347389 A330774 A296307

Adjacent sequences: A356772 A356773 A356774 * A356776 A356777 A356778

Keyword

sign

Author

Paul D. Hanna, Sep 22 2022

Status

approved