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A143323 Expansion of eta(q^2)^4 * eta(q^5) * eta(q^20)^2 / ( eta(q) * eta(q^4)^2 * eta(q^10)^2 ) in powers of q. 0
1, 1, -2, -1, 1, -2, -2, 1, 3, 1, 0, 2, 0, -2, -2, -1, 0, 3, 0, -1, 4, 0, -2, -2, 1, 0, -4, 2, 2, -2, 0, 1, 0, 0, -2, -3, 0, 0, 0, 1, 2, 4, -2, 0, 3, -2, -2, 2, 3, 1, 0, 0, 0, -4, 0, -2, 0, 2, 0, 2, 2, 0, -6, -1, 0, 0, -2, 0, 4, -2, 0, 3, 0, 0, -2, 0, 0, 0, 0, -1, 5, 2, -2, -4, 0, -2, -4, 0, 2, 3, 0, 2, 0, -2, 0, -2, 0, 3, 0, -1, 2, 0, -2, 0, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Ramanujan theta functions: f(q) := Prod_{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) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

REFERENCES

L.-C. Shen, On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5. Trans. Amer. Math. Soc. 345 (1994), no. 1, 323-345. See p. 338, Eq. (3.22), p. 342, Eq. (3.41).

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 q * phi(-q^2) * chi(q) * psi(q^10) * chi(-q^5) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Euler transform of period 20 sequence [ 1, -3, 1, -1, 0, -3, 1, -1, 1, -2, 1, -1, 1, -3, 0, -1, 1, -3, 1, -2, ...].

Multiplicative with a(2^e) = -(-1)^e unless e=0, a(p^e) = 1 if p=5, a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e+1 if p == 1, 9 (mod 20), a(p^e) = (e+1)(-1)^e if p == 3, 7 (mod 20).

G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A129391.

G.f.: Sum_{k>0} -(-1)^k F(x^(2*k - 1)) where F(x) = x * (1 + x) * (1 - x^2) / (1 + x^5).

G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2 * (1 - x^(5*k)) * ( 1 + x^(10*k))^2.

EXAMPLE

q + q^2 - 2*q^3 - q^4 + q^5 - 2*q^6 - 2*q^7 + q^8 + 3*q^9 + q^10 + ...

PROG

(PARI) {a(n) = if( n<1, 0, (-1)^n * (qfrep([2, 1; 1, 3], n)[n] -qfrep([1, 0; 0, 5], n)[n] ))}

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^5 + A) * eta(x^20 + A)^2 / ( eta(x + A) * eta(x^4 + A)^2 * eta(x^10 + A)^2 ), n))}

CROSSREFS

-(-1)^n * A111494(n) = a(n).

Sequence in context: A124233 A035170 A111949 * A086598 A211261 A074746

Adjacent sequences:  A143320 A143321 A143322 * A143324 A143325 A143326

KEYWORD

sign,mult

AUTHOR

Michael Somos, Aug 07 2008

STATUS

approved

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Last modified November 22 10:43 EST 2014. Contains 249805 sequences.