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A132004 Expansion of (1 - phi(q^3)/ phi(q) * phi(-q^2) * phi(-q^6)) / 2 in powers of q where phi() is a Ramanujan theta function. 2
1, -1, 1, -1, 2, -1, 0, -1, 1, -2, 0, -1, 2, 0, 2, -1, 2, -1, 0, -2, 0, 0, 0, -1, 3, -2, 1, 0, 2, -2, 0, -1, 0, -2, 0, -1, 2, 0, 2, -2, 2, 0, 0, 0, 2, 0, 0, -1, 1, -3, 2, -2, 2, -1, 0, 0, 0, -2, 0, -2, 2, 0, 0, -1, 4, 0, 0, -2, 0, 0, 0, -1, 2, -2, 3, 0, 0, -2, 0, -2, 1, -2, 0, 0, 4, 0, 2, 0, 2, -2, 0, 0, 0, 0, 0, -1, 2, -1, 0, -3, 2, -2, 0, -2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

REFERENCES

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.72).

LINKS

Table of n, a(n) for n=1..105.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (1 - eta(q)^2* eta(q^4)* eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3)) / 2 in powers of q.

a(n) is multiplicative with b(2^e) = 2*0^e -1, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).

G.f.: Sum_{k>0} x^k / (1 + x^k) * kronecker( -36, k).

a(3*n) = a(n). -2 * a(n) = A132003(n) unless n=0. -a(2*n) = A035154(n). a(2*n + 1) = A125079(n).

EXAMPLE

x - x^2 + x^3 - x^4 + 2*x^5 - x^6 - x^8 + x^9 - 2*x^10 - x^12 + 2*x^13 + ...

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(n+d) * kronecker( -36, d)))}

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3)) / 2, n))}

(PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==3, 1, if( p==2, -1, if( p%4==1, e+1, !(e%2)))))))}

CROSSREFS

Cf. A035154, A125079, A132003.

Sequence in context: A035154 A113446 A121450 * A143110 A109294 A132966

Adjacent sequences:  A132001 A132002 A132003 * A132005 A132006 A132007

KEYWORD

sign,mult

AUTHOR

Michael Somos, Aug 06 2007

STATUS

approved

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Last modified November 27 01:09 EST 2014. Contains 250152 sequences.