A184363 G.f.: eta(x)^3*(1 + x*eta'(x)/eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

1, -4, 0, 10, 0, 0, -21, 0, 0, 0, 39, 0, 0, 0, 0, -66, 0, 0, 0, 0, 0, 104, 0, 0, 0, 0, 0, 0, -155, 0, 0, 0, 0, 0, 0, 0, 221, 0, 0, 0, 0, 0, 0, 0, 0, -304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 406, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -529, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 675, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -846

Offset

0,2

Links

Table of n, a(n) for n=0..91.

Formula

G.f.: A(x) = Sum_{n>=0} (-1)^n*(2n+1)*(n^2+n+6)/6 * x^(n(n+1)/2).

G.f.: A(x) = eta(x)^2*G(x) where G(x) is the g.f. of A184362.

Example

G.f.: A(x) = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 +...
A(x) = eta(x)^3*[1 + x*d/dx log(eta(x))] where
eta(x)^3 = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 +...+ (-1)^n*(2n+1)*x^(n(n+1)/2) +...
1 + x*d/dx log(eta(x)) = 1 - x - 3*x^2 - 4*x^3 - 7*x^4 - 6*x^5 - 12*x^6 - 8*x^7 - 15*x^8 +...+ -sigma(n)*x^n +...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, (-1)^m*(2*m+1)*(m^2+m+6)/6*x^(m*(m+1)/2)), n)}
(PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^3*(1+x*deriv(log(eta(x+x*O(x^n))))), n)}

Crossrefs

Cf. A184362, A184366.

Sequence in context: A330386 A098487 A174381 * A331451 A164735 A293933

Adjacent sequences: A184360 A184361 A184362 * A184364 A184365 A184366

Keyword

sign

Author

Paul D. Hanna, Jan 18 2011

Status

approved