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A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.

%I A262146 #22 Feb 16 2025 08:33:27
%S A262146 1,2,4,8,15,25,42,68,107,166,253,377,557,811,1166,1661,2344,3275,4543,
%T A262146 6253,8544,11600,15653,20994,28011,37178,49100,64550,84489,110115,
%U A262146 142951,184867,238196,305844,391391,499244,634865,804925,1017610,1282957,1613195
%N A262146 Expansion of f(-x, -x^5) * f(x, x^7) / f(-x, -x^2)^2 in powers of x where f(, ) is Ramanujan's general theta function.
%C A262146 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A262146 G. C. Greubel, <a href="/A262146/b262146.txt">Table of n, a(n) for n = 0..1000</a>
%H A262146 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A262146 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A262146 Expansion of - (psi(x^6) / psi(x) - psi(x^6) / psi(-x)) / (2 * x) in powers of x^2 where psi() is a Ramanujan theta function.
%F A262146 Euler transform of period 48 sequence [ 2, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 2, 2, 1, 0, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 2, 1, 2, 0, ...].
%F A262146 a(n) = A132217(2*n + 1) = - A262160(2*n + 1).
%F A262146 Convolution product of A097451 and A078408.
%F A262146 a(n) ~ exp(Pi*sqrt(n)) / (2^(7/2) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Mar 31 2018
%e A262146 G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 15*x^4 + 25*x^5 + 42*x^6 + 68*x^7 + ...
%e A262146 G.f. = q^13 + 2*q^29 + 4*q^45 + 8*q^61 + 15*q^77 + 25*q^93 + 42*q^109 + ...
%t A262146 a[ n_] := SeriesCoefficient[ - x^(-5/8) EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, 2 n + 1}];
%o A262146 (PARI) {a(n) = my(A); if( n<0, 0, n = 2*n + 1; A = x * O(x^n); polcoeff( - eta(x + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^2 * eta(x^6 + A)), n))};
%Y A262146 Cf. A078408, A097451, A132217, A262160.
%K A262146 nonn
%O A262146 0,2
%A A262146 _Michael Somos_, Oct 06 2015