Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Mar 12 2021 22:24:48
%S 1,1,1,2,3,4,6,8,11,14,18,24,30,38,48,60,75,92,114,140,170,208,252,
%T 304,366,439,526,626,744,884,1044,1232,1451,1704,1998,2336,2730,3182,
%U 3700,4300,4986,5772,6672,7700,8876,10212,11736,13472,15438,17673,20207
%N Expansion of psi(q^6) * f(-q^12) / (psi(-q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Vaclav Kotesovec, <a href="/A261154/b261154.txt">Table of n, a(n) for n = 0..2000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of eta(q^2) * eta(q^9) * eta(q^12)^3 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)^2) in powers of q.
%F Euler transform of period 36 sequence [1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, -1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A186115.
%F a(n) = A233693(n) unless n=0. a(2*n) = A212484(n).
%F a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + ...
%t a[ n_] := SeriesCoefficient[ 2^(1/2) q^(1/2) EllipticTheta[ 2, 0, q^3] QPochhammer[ q^12] / (EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)^2), n))};
%Y Cf. A186115, A212484, A233693.
%K nonn
%O 0,4
%A _Michael Somos_, Aug 10 2015