A029845 Expansion of 16/lambda(z) in powers of nome q = exp(Pi*i*z).

1, 8, 20, 0, -62, 0, 216, 0, -641, 0, 1636, 0, -3778, 0, 8248, 0, -17277, 0, 34664, 0, -66878, 0, 125312, 0, -229252, 0, 409676, 0, -716420, 0, 1230328, 0, -2079227, 0, 3460416, 0, -5677816, 0, 9198424, 0, -14729608, 0, 23328520, 0, -36567242, 0, 56774712, 0

Offset

-1,2

Comments

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (terms -1..1000 from Alois P. Heinz)

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Elliptic Lambda Function

Formula

Expansion of (eta(q^2)^3/(eta(q)*eta(q^4)^2))^8 in powers of q. - Michael Somos, Nov 14 2006

Expansion of (chi(q)*chi(-q^2))^8/q in powers of q where chi() is a Ramanujan theta function.

Euler transform of period 4 sequence [ 8, -16, 8, 0, ...]. - Michael Somos, Nov 14 2006

G.f. A(x) satisfies: 0=f(A(x), A(x^2)) where f(u, v) = 256 - v*(32-16*u+u^2) + v^2. - Michael Somos, Nov 14 2006

G.f.: 1/q*(Product_{k>0} (1+q^(2k-1))/(1+q^(2k)))^8.

Example

1/q + 8 + 20*q - 62*q^3 + 216*q^5 - 641*q^7 + 1636*q^9 - 3778*q^11 + ...

Mathematica

QP = QPochhammer; s = 16*q + (QP[q]/QP[q^4])^8 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

Prog

(PARI) {a(n)=local(A); if(n<-1, 0, n++; A=x*O(x^n); polcoeff( 16*x+(eta(x+A)/eta(x^4+A))^8, n))} /* Michael Somos, Nov 14 2006 */

Crossrefs

Cf. A007248(n) = a(2n-1). A124972(n) = a(n) except at n=0.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A153704 A344052 A316201 * A124972 A161969 A000731

Adjacent sequences: A029842 A029843 A029844 * A029846 A029847 A029848

Keyword

sign,easy,changed

Author

N. J. A. Sloane

Status

approved