A356774 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^n * (1 - x^n)^(n-2).

1, 4, 7, 11, 16, 17, 29, 21, 46, 21, 67, 22, 92, 1, 151, -23, 154, 22, 191, -118, 407, -175, 277, 23, 326, -363, 946, -643, 436, 282, 497, -1199, 1948, -1019, 701, -47, 704, -1519, 3641, -3127, 862, 1759, 947, -5301, 7036, -2943, 1129, -1187, 1226, -2149, 10252

Offset

1,2

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*(n-1)/2 * x^n * (1 - x^n)^(n-2).

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..2050

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

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

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

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

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

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

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

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

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

Example

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 + ...
where
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) + ...

Prog

(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) )) );
polcoeff(A, n)}
for(n=1, 100, print1(a(n), ", "))

Crossrefs

Cf. A291937, A356775, A357156, A357157.

Sequence in context: A310741 A078916 A236771 * A310742 A310743 A310744

Adjacent sequences: A356771 A356772 A356773 * A356775 A356776 A356777

Keyword

sign

Author

Paul D. Hanna, Sep 22 2022

Status

approved