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A131124 Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where phi(), psi() are Ramanujan theta functions. 3
1, -4, 4, 0, 2, 0, -8, 0, -1, 0, 20, 0, -2, 0, -40, 0, 3, 0, 72, 0, 2, 0, -128, 0, -4, 0, 220, 0, -4, 0, -360, 0, 5, 0, 576, 0, 8, 0, -904, 0, -8, 0, 1384, 0, -10, 0, -2088, 0, 11, 0, 3108, 0, 12, 0, -4552, 0, -15, 0, 6592, 0, -18, 0, -9448, 0, 22, 0, 13392 (list; graph; refs; listen; history; text; internal format)
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 3 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 21 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(8). [Yang 2004] - Michael Somos, Jul 21 2014
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
FORMULA
Expansion of (eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2 ))^2 in powers of q.
Euler transform of period 8 sequence [ -4, -2, -4, -4, -4, -2, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107035.
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.
G.f.: (x * Product_{k>0} (1 + x^k)^4 * (1 + x^(2*k))^2 * (1 + x^(4*k))^4 )^-1.
Convolution inverse of A107035.
a(2*n) = 0 unless n=0. a(n) = A029841(n) unless n=0. a(4*n - 1) = A029839(n). a(4*n + 1) = 4 * A079006(n).
EXAMPLE
G.f. = 1/q - 4 + 4*q + 2*q^3 - 8*q^5 - q^7 + 20*q^9 - 2*q^11 - 40*q^13 + 3*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 4 (EllipticTheta[ 4, 0, q] / EllipticTheta[ 2, 0, q^2])^2, {q, 0, n}]; (* Michael Somos, Apr 24 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) / (eta(x^2 + A) * eta(x^8 + A)^2) )^2, n))};
CROSSREFS
Sequence in context: A108068 A283998 A318143 * A131125 A187149 A106508
KEYWORD
sign
AUTHOR
Michael Somos, Jun 15 2007
STATUS
approved

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)