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 #19 Oct 09 2023 11:48:14
%S 1,1,4,5,9,13,21,29,46,62,90,122,171,227,311,408,545,709,933,1198,
%T 1555,1980,2536,3205,4063,5092,6400,7966,9928,12281,15198,18684,22979,
%U 28097,34346,41789,50813,61527,74453,89757,108114,129809,155704,186221,222503
%N Expansion of phi(x^2) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A226622/b226622.txt">Table of n, a(n) for n = 0..1000</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 q^(1/24) * eta(q^4)^5 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.
%F Euler transform of period 8 sequence [ 1, 3, 1, -2, 1, 3, 1, 0, ...].
%F G.f.: (Sum_{k in Z} x^(2*k^2)) / (Product_{k>0} (1 - x^k)).
%F a(n) = A022597(2*n) = A073252(2*n).
%F G.f. A(x) satisfies A(x^2) = ( chi(x)^2 + chi(-x)^2 )/2, where chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700. Cf. A226635. - _Peter Bala_, Sep 29 2023
%e 1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 13*x^5 + 21*x^6 + 29*x^7 + 46*x^8 + 62*x^9 + ...
%e 1/q + q^23 + 4*q^47 + 5*q^71 + 9*q^95 + 13*q^119 + 21*q^143 + 29*q^167 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] / QPochhammer[ q], {q, 0, n}]
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))}
%Y Cf. A000700, A022597, A073252, A226635.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Aug 31 2013