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A260145 Expansion of x * (psi(x^4) / phi(x))^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A260145 #11 Feb 16 2025 08:33:26
%S A260145 1,-4,12,-32,78,-176,376,-768,1509,-2872,5316,-9600,16966,-29408,
%T A260145 50088,-83968,138738,-226196,364284,-580032,913824,-1425552,2203368,
%U A260145 -3376128,5130999,-7738136,11585208,-17225472,25444278,-37350816,54504160,-79085568,114133296
%N A260145 Expansion of x * (psi(x^4) / phi(x))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A260145 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A260145 G. C. Greubel, <a href="/A260145/b260145.txt">Table of n, a(n) for n = 1..1000</a>
%H A260145 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A260145 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A260145 Expansion of (eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5)^2 in powers of q.
%F A260145 Euler transform of period 8 sequence [ -4, 6, -4, 4, -4, 6, -4, 0, ...].
%F A260145 G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A210067.
%F A260145 G.f.: x * Product_{k>0} ( 1 + x^(2*k))^6 * (1 + x^(4*k))^4 / (1 + x^k)^4.
%F A260145 a(n) = -(-1)^n * A107035(n). -4 * a(n) = A210066(n) unless n=0. -8 * a(n) = A139820(n) unless n=0.
%F A260145 a(2*n) = -4 * A092877(n). a(2*n + 1) = A022577(n). a(4*n) = -32 * A014103(n).
%F A260145 Convolution square of A210063. Convolution inverse of A131125.
%F A260145 a(n) ~ -(-1)^n * exp(sqrt(2*n)*Pi) / (64 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 17 2017
%e A260145 G.f. = x - 4*x^2 + 12*x^3 - 32*x^4 + 78*x^5 - 176*x^6 + 376*x^7 + ...
%t A260145 a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^2]^2 / EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
%o A260145 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5)^2, n))};
%Y A260145 Cf. A014103, A022577, A092877, A107035, A131125, A139820, A210063, A210066, A210067.
%K A260145 sign
%O A260145 1,2
%A A260145 _Michael Somos_, Jul 17 2015