A357156 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).

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, -923, 1, 1277, 1, -1727, 1002, 1, 1681, -2021, 1, 1, 1821, -1027, 1, 912, 1, -7721, 11146, 1, 1, -12571, 736, 15401, 4846, -17016, 1, -6389, 27457, -20956, 7316, 1, 1, -6486, 1, 1, 22177

Offset

3,4

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)*(n+2)/24 * x^(3*n) * (1 - x^n)^(n-2).

(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).

Links

Paul D. Hanna, Table of n, a(n) for n = 3..2050

Formula

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

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

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

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

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

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

(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).

Example

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 + ...
where
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) + ...

Prog

(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) )) );
polcoeff(A, n)}
for(n=3, 100, print1(a(n), ", "))

Crossrefs

Cf. A291937, A356774, A356775, A357157.

Sequence in context: A376730 A109001 A203005 * A296963 A176560 A152602

Adjacent sequences: A357153 A357154 A357155 * A357157 A357158 A357159

Keyword

sign

Author

Paul D. Hanna, Sep 22 2022

Status

approved