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A145707 Expansion of chi(-q) / chi(-q^10) in powers of q where chi() is a Ramanujan theta function.

%I A145707 #20 Feb 16 2025 08:33:09
%S A145707 1,-1,0,-1,1,-1,1,-1,2,-2,3,-3,3,-4,4,-5,6,-6,7,-8,10,-11,11,-13,15,
%T A145707 -17,18,-20,23,-25,29,-32,34,-39,42,-47,52,-56,62,-68,77,-83,89,-99,
%U A145707 108,-119,129,-139,154,-167,183,-199,214,-234,253,-276,299,-322,350
%N A145707 Expansion of chi(-q) / chi(-q^10) in powers of q where chi() is a Ramanujan theta function.
%C A145707 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A145707 In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - _Vaclav Kotesovec_, Aug 31 2015
%H A145707 Alois P. Heinz, <a href="/A145707/b145707.txt">Table of n, a(n) for n = 0..10000</a>
%H A145707 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A145707 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A145707 Expansion of q^(-3/8) * eta(q) * eta(q^20) / (eta(q^2) * eta(q^10)) in powers of q.
%F A145707 Euler transform of period 20 sequence [ -1, 0, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, ...].
%F A145707 G.f. is a period 1 Fourier series which satisfies f(-1 / (1280 t)) = f(t) where q = exp(2 Pi i t).
%F A145707 G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(20*k - 10)).
%F A145707 a(n) = (-1)^n * A145703(n) = A145704(2*n + 1) = - A145705(2*n + 1).
%F A145707 a(n) ~ (-1)^n * exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015
%e A145707 G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 3*x^10 + ...
%e A145707 G.f. = q^3 - q^11 - q^27 + q^35 - q^43 + q^51 - q^59 + 2*q^67 - 2*q^75 + ...
%t A145707 nmax = 100; CoefficientList[Series[Product[(1 + x^(10*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 30 2015 *)
%t A145707 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)
%o A145707 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^10 + A)), n))};
%Y A145707 Cf. A145703, A145704, A145705.
%Y A145707 Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9).
%K A145707 sign
%O A145707 0,9
%A A145707 _Michael Somos_, Oct 17 2008