A185654 G.f.: exp( Sum_{n>=1} -sigma(3n)*x^n/n ).

1, -4, 2, 9, -9, -2, 0, -5, 9, 9, 0, -9, -1, -9, 0, -1, 9, 9, -9, 9, 0, 9, -5, -18, -18, 9, 7, 0, 9, 0, 0, 9, 9, -18, 18, -7, -9, -9, -9, 9, -4, -9, -9, 18, 9, 0, 18, 9, 0, -9, -9, -8, -9, 18, -9, 9, -18, 1, -9, -18, 9, 0, 18, 18, 0, 0, 9, -9, 18, -9, 5, -9, 0, -9, -9, -9, -18, 11, 9

Offset: 0

Author

Paul D. Hanna, Feb 16 2011

Keywords

• sign

Examples

			G.f. = 1 - 4*x + 2*x^2 + 9*x^3 - 9*x^4 - 2*x^5 - 5*x^7 + 9*x^8 + ... - _Michael Somos_, Jul 12 2018
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A182819, A144613, A000203.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), this sequence (k=3), A282937 (k=5), A282942 (k=7).

Programs

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4 / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jul 12 2018 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n,-sigma(3*m)*x^m/m)+x*O(x^n)),n)}

PARI

{a(n)=local(X=x+x*O(x^n));polcoeff(eta(X)^4/eta(X^3),n)}

Formula

G.f.: E(x)^4/E(x^3) where E(x) = Product_{n>=1} (1-x^n). [From a formula by Joerg Arndt in A182819]

a(n) = -(1/n)*Sum_{k=1..n} sigma(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

Expansion of E(x) * E(x*w) * E(x/w) in powers of x^3 where w = exp(2 Pi i / 3). - Michael Somos, Jul 12 2018