A184365 G.f.: eta(x) - x*eta'(x), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

1, 0, 1, 0, 0, -4, 0, -6, 0, 0, 0, 0, 11, 0, 0, 14, 0, 0, 0, 0, 0, 0, -21, 0, 0, 0, -25, 0, 0, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 39, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -50, 0, 0, 0, 0, 0, -56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 69, 0, 0, 0, 0, 0, 0, 76, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -91, 0, 0, 0

Offset

0,6

Links

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

Formula

G.f. satisfies: [x^(n+1)] A(x)*eta(x)^n = 0 for n>=0.

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

G.f.: A(x) = G(x)/eta(x)^2 where G(x) = Sum_{n>=0} -(-1)^n*(n-2)(n+3)(2n+1)/6*x^(n(n+1)/2) is the g.f. of A184366.

Example

G.f.: A(x) = 1 + x^2 - 4*x^5 - 6*x^7 + 11*x^12 + 14*x^15 - 21*x^22 - 25*x^26 + 34*x^35 + 39*x^40 - 50*x^51 +...
Illustrate the property: [x^(n+1)] A(x)*eta(x)^n = 0
in the table of coefficients of A(x)*eta(x)^n for n=0..10:
[1,(0), 1, 0, 0, -4, 0, -6, 0, 0, 0, 0, 11, 0, 0, 14,...];
[1, -1,(0), -1, -1, -3, 4, 0, 6, 7, -4, 0, 0, -11,...];
[1, -2, 0,(0), 0, 0, 7, 0, 0, 0, -21, 0, 0, 0, 0, 44,...];
[1, -3, 1, 2,(0), 1, 5, -6, -9, 0, -21, 28, 20, 9,...];
[1, -4, 3, 4, -3,(0), 1, -10, -9, 16, -9, 54, 7, -40,...];
[1, -5, 6, 5, -10, 0,(0), -7, 0, 35, -12, 45, -49, -105,...];
[1, -6, 10, 4, -21, 6, 5,(0), 7, 38, -42, 12, -90, -96,...];
[1, -7, 15, 0, -35, 24, 14,0,(0), 20, -77, 0, -55, 0,...];
[1, -8, 21, -8, -50, 60, 18,-22,-21,(0), -73, 36, 45, 76,...];
[1, -9, 28, -21, -63, 119, 0, -78,-33,14,(0), 77, 119, 0,...];
[1, -10, 36, -40, -70, 204, -65, -168,15,90,117,(0),...]; ...
so that the coefficient of x^(n+1) in A(x)*eta(x)^n is zero for n>=0.
Note: the g.f.s of the diagonals in the above table are powers of G(x), where G(x) = eta(x*G(x)) is the g.f. of A066398.
The g.f. of A184366 equals:
A(x)*eta(x)^2 = 1 - 2*x + 7*x^6 - 21*x^10 + 44*x^15 - 78*x^21 + 125*x^28 - 187*x^36 +...+ -(-1)^n*(n-2)(n+3)(2n+1)/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. A184366, A184362, A066398.

Sequence in context: A134530 A351571 A308678 * A126813 A056141 A246004

Adjacent sequences: A184362 A184363 A184364 * A184366 A184367 A184368

Keyword

sign

Author

Paul D. Hanna, Jan 17 2011

Status

approved