A184362 G.f.: eta(x) + x*eta'(x).

1, -2, -3, 0, 0, 6, 0, 8, 0, 0, 0, 0, -13, 0, 0, -16, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0, -36, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -71, 0, 0, 0, 0, 0, 0, -78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 93, 0, 0, 0

Offset

0,2

Comments

The formulas specified in this entry use eta(x) to denote Dedekind's eta(q) function without the q^(1/24) factor.

Links

Table of n, a(n) for n=0..95.

Formula

G.f. A(x) satisfies:

(1) [x^n] A(x)/eta(x)^(n+1) = 0 for n>=1.

(2) [x^n] A(x)/eta(x)^n = A109084(n) for n>=0.

(3) [x^n] A(x)/eta(x)^(n+2) = A109085(n) for n>=0.

(4) A(x)/eta(x) = 1 - Sum_{n>=1} sigma(n)*x^n.

(5) A(x) = 1 + Sum_{n>=1} (-1)^n*[n(3n-1)/2+1 + (n(3n+1)/2+1)*x^n)] * x^(n(3n-1)/2).

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

Example

G.f.: A(x) = 1 - 2*x - 3*x^2 + 6*x^5 + 8*x^7 - 13*x^12 - 16*x^15 + 23*x^22 + 27*x^26 - 36*x^35 - 41*x^40 +...
Illustrate the property: [x^n] A(x)/eta(x)^(n+1) = 0
in the table of coefficients of A(x)/eta(x)^(n+1) for n=0..10:
[1, -1, -3, -4, -7, -6, -12, -8, -15, -13, -18,...,-sigma(n),...];
[1,(0), -2, -6, -15, -28, -55, -90, -154, -240, -378,...];
[1, 1,(0), -5, -20, -54, -130, -275, -555, -1050, -1924,...];
[1, 2, 3,(0), -17, -72, -221, -572, -1350, -2958, -6160,...];
[1, 3, 7, 10,(0), -63, -287, -930, -2580, -6475, -15162,...];
[1, 4, 12, 26, 38,(0), -253, -1196, -4059, -11780, -31027,...];
[1, 5, 18, 49, 105, 153,(0), -1062, -5175, -18140, -54544,...];
[1, 6, 25, 80, 210, 442, 646,(0), -4615, -22990, -82671,...];
[1, 7, 33, 120, 363, 924, 1926, 2816,(0), -20570, -104285,...];
[1, 8, 42, 170, 575, 1668, 4161, 8602, 12585,(0), -93538,...];
[1, 9, 52, 231, 858, 2756, 7766, 19071, 39182, 57343,(0),...]; ...
so that the coefficient of x^n in A(x)/eta(x)^(n+1) is zero for n>=1.
Note: the g.f.s of the diagonals in the above table are powers of G(x),
where G(x) = 1/eta(x*G(x)) is the g.f. of A109085.
The g.f. of A184363 equals:
A(x)*eta(x)^2 = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 +...+ (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2) +...

Prog

(PARI) {a(n)=polcoeff(eta(x+x*O(x^n)) + x*deriv(eta(x+x*O(x^n))), n)}

Crossrefs

Cf. A184363, A184365, A109084, A109085, A000203.

Sequence in context: A368243 A259657 A332672 * A354443 A011311 A240658

Adjacent sequences: A184359 A184360 A184361 * A184363 A184364 A184365

Keyword

sign

Author

Paul D. Hanna, Jan 18 2011

Extensions

Example of g.f. corrected by Paul D. Hanna, Jan 18 2011

Name changed slightly by Paul D. Hanna, Nov 27 2012

Status

approved