OFFSET
0,4
COMMENTS
REFERENCES
B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 345 Entry 1(i).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 155 Eq. (9.13)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of chi(-q) / chi(-q^3)^3 where chi() is a Ramanujan theta function.
Expansion of q^(-1/3) * c(q^2) / c(q) where c() is a cubic AGM theta function. - Michael Somos, Oct 04 2006
Expansion of q^(-1/3) * eta(q) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [-1, 0, 2, 0, -1, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + 2*u*v^2.
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^3 - v^2 + v) - u^3 * (1 + 2*v + 4*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A141094. - Michael Somos, Dec 07 2013
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3.
G.f.: 1 / (1 + (x + x^2) / (1 + (x^2 + x^4) / (1 + (x^3 + x^6) / ...))).
G.f.: T(0), where T(k) = 1 - (x^(k+1)+x^(2*k+2))/((x^(k+1)+x^(2*k+2))+1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013
EXAMPLE
G.f. = 1 - x + 2*x^3 - 2*x^4 - x^5 + 4*x^6 - 4*x^7 - x^8 + 8*x^9 + ...
G.f. = q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}] / Product[ 1 - x^k, {k, 3, n, 6}]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] / QPochhammer[ x^3, x^6]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3), n))}; /* Michael Somos, Dec 07 2013 */
(PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A + (x*A^2)^2) - x*A^2); polcoeff(A, n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=0, (n-1)\2, (1 - x^(2*k + 1))^if(k%3==1, -2, 1), 1 + x * O(x^n)), n))};
CROSSREFS
KEYWORD
sign,changed
AUTHOR
Michael Somos, Mar 07 2004
STATUS
approved