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A226622 Expansion of phi(x^2) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

%I A226622 #22 Jun 29 2025 07:26:48
%S A226622 1,1,4,5,9,13,21,29,46,62,90,122,171,227,311,408,545,709,933,1198,
%T A226622 1555,1980,2536,3205,4063,5092,6400,7966,9928,12281,15198,18684,22979,
%U A226622 28097,34346,41789,50813,61527,74453,89757,108114,129809,155704,186221,222503
%N A226622 Expansion of phi(x^2) / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.
%C A226622 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A226622 G. C. Greubel, <a href="/A226622/b226622.txt">Table of n, a(n) for n = 0..1000</a>
%H A226622 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226622 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A226622 Expansion of q^(1/24) * eta(q^4)^5 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.
%F A226622 Euler transform of period 8 sequence [ 1, 3, 1, -2, 1, 3, 1, 0, ...].
%F A226622 G.f.: (Sum_{k in Z} x^(2*k^2)) / (Product_{k>0} (1 - x^k)).
%F A226622 a(n) = A022597(2*n) = A073252(2*n).
%F A226622 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
%F A226622 a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 29 2025
%e A226622 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 A226622 1/q + q^23 + 4*q^47 + 5*q^71 + 9*q^95 + 13*q^119 + 21*q^143 + 29*q^167 + ...
%t A226622 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] / QPochhammer[ q], {q, 0, n}]
%o A226622 (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 A226622 Cf. A000700, A022597, A073252, A226635.
%K A226622 nonn,easy
%O A226622 0,3
%A A226622 _Michael Somos_, Aug 31 2013