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A139139 Expansion of (phi(q) / phi(q^3) - 1) / 2 in powers of q where phi() is a Ramanujan theta function. 2
1, 0, -1, -1, 0, 2, 2, 0, -3, -4, 0, 5, 6, 0, -8, -9, 0, 12, 14, 0, -18, -20, 0, 26, 29, 0, -37, -42, 0, 52, 58, 0, -72, -80, 0, 99, 110, 0, -134, -148, 0, 180, 198, 0, -240, -264, 0, 317, 347, 0, -416, -454, 0, 542, 592, 0, -702, -764, 0, 904, 982, 0, -1158, -1257, 0, 1476, 1598, 0, -1872, -2024, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

LINKS

Table of n, a(n) for n=1..71.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q * chi(-q^2) * psi(q^6)^2 / (psi(q^3) * f(-q^5, -q^7)) in powers of q where phi(), f() are Ramanujan theta functions.

Euler transform of period 12 sequence [ 0, -1, -1, 0, 1, 2, 1, 0, -1, -1, 0, 0, ...].

a(3*n + 2) = 0.

G.f.: ((Sum_k x^k^2) / (Sum_k x^(3*k^2)) - 1) / 2

G.f.: Product_{k>0} (1 + x^(2*k))^2 * (1 - x^(2*k) + x^(4*k))^3 / ( (1 + x^k) * (1 - x^k + x^(2*k)) * (1 - x^(12*k - 5)) * (1 - x^(12*k - 7))).

EXAMPLE

q - q^3 - q^4 + 2*q^6 + 2*q^7 - 3*q^9 - 4*q^10 + 5*q^12 + 6*q^13 + ...

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5) - 1) / 2, n))}

CROSSREFS

Cf. A139137(n) = 2 * a(n) unless n=0.

Sequence in context: A207383 A191362 A137422 * A077872 A239292 A094053

Adjacent sequences:  A139136 A139137 A139138 * A139140 A139141 A139142

KEYWORD

sign

AUTHOR

Michael Somos, Apr 10 2008

STATUS

approved

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Last modified August 28 08:35 EDT 2015. Contains 261118 sequences.