OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of psi(x)^4 - 9 * x * psi(x^3)^4 in powers of x where psi() is a Ramanujan theta function.
Expansion of x^(-1/2) * (b(x)^3 * c(x^2)^2 / (3 * c(x)))^(1/2) in powers of x where b(), c() are cubic AGM functions.
Expansion of q^(-1/2) * eta(q)^5 * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [-5, -4, -2, -4, -5, -4, ...].
a(n) = b(2*n+1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1)/(p - 1) if p>5.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 18 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124449
G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^3.
G.f.: Sum_{k>0} k * f(x^k) - 9 * k * f(x^(3*k)) where f(x) = x * (1 - x) / ((1 + x) * (1 + x^2)).
G.f.: f(x) - 3 * f(x^2) - 9 * f(x^3) + 2 * f(x^4) + 27 * f(x^6) - 18 * f(x^12) where f() is the g.f. of A000203.
EXAMPLE
G.f. = 1 - 5*x + 6*x^2 + 8*x^3 - 23*x^4 + 12*x^5 + 14*x^6 - 30*x^7 + 18*x^8 + ...
G.f. = q - 5*q^3 + 6*q^5 + 8*q^7 - 23*q^9 + 12*q^11 + 14*q^13 - 30*q^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/2) x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 4, 0, x]^3 QPochhammer[ -x^3, x^3]^3, {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
a[ n_] := SeriesCoefficient[ (1/16) x^(-1/2) (EllipticTheta[ 2, 0, x^(1/2)]^4 - 9 EllipticTheta[ 2, 0, x^(3/2)]^4), {x, 0, n}]; (* Michael Somos, Oct 27 2015 *)
PROG
(PARI) {a(n) = my(A); if ( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^5 * eta(x^6 + A)^3 / ( eta(x^2 + A) * eta(x^3 + A)^3 ), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 06 2007
STATUS
approved