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 q^(1/24) * eta(q^2)^5 /(eta(q)^3 * eta(q^4)^2) in powers of q. - Michael Somos, Sep 17 2004
Euler transform of period 4 sequence [3, -2, 3, 0, ...]. - Michael Somos, Sep 17 2004
G.f. A(x) is the limit of x^(n^2) P_{2n}(1/x) where P_n(q) = Sum_{k=0..n} C(n,k;q) and C(n,k;q) is q-binomial coefficients. See A083906 for P_n. - Michael Somos, Sep 17 2004
G.f.: (1 + 2 * Sum_{k>0} x^(k^2)) / (Product_{k>0} (1 - x^k)).
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(7/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, May 01 2017
Expansion of chi(x)^3/chi(-x^2) = chi(x)^2/chi(-x) = chi(-x^2)^2/chi(-x)^3 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Apr 24 2023
EXAMPLE
G.f. = 1 + 3*x + 4*x^2 + 7*x^3 + 13*x^4 + 19*x^5 + 29*x^6 + 43*x^7 + ...
G.f. = 1/q + 3*q^23 + 4*q^47 + 7*q^71 + 13*q^95 + 19*q^119 + 29*q^143 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Oct 29 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^2 / QPochhammer[ q, q^2], {q, 0, n}]; (* Michael Somos, Oct 29 2013 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1) / eta(x + x * O(x^n)), n))}; /* Michael Somos, Sep 17 2004 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^2), n)); } /* Michael Somos, Sep 17 2004 */
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, 2*n, prod(i=1, k, (1 -x^(2*n + 1-i)) / (1 - x^i))), n^2-n))}; /* Michael Somos, Sep 17 2004 */
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved