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A121361 Expansion of f(x^1, x^5) * psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

%I A121361 #16 Feb 16 2025 08:33:02
%S A121361 1,1,1,1,0,1,1,2,1,0,1,1,1,1,1,0,1,1,1,0,2,2,1,1,0,1,0,1,2,0,1,1,0,2,
%T A121361 0,2,1,0,1,1,1,1,2,1,0,1,2,1,0,0,1,1,1,1,0,0,2,1,2,0,1,1,1,2,1,1,0,1,
%U A121361 1,0,1,1,2,1,0,1,1,3,0,0,1,0,1,0,0,2,1,1,1,1,1,2,0,1,0,2,2,1,3,0,0,0,1,0,0
%N A121361 Expansion of f(x^1, x^5) * psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.
%C A121361 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A121361 G. C. Greubel, <a href="/A121361/b121361.txt">Table of n, a(n) for n = 0..1000</a>
%H A121361 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A121361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A121361 Expansion of q^(-7/12) * eta(q^2) * eta(q^3) * eta(q^4) * eta(q^12) /
%F A121361 (eta(q) * eta(q^6)) in powers of q.
%F A121361 Euler transform of period 12 sequence [ 1, 0, 0, -1, 1, 0, 1, -1, 0, 0, 1, -2, ...].
%F A121361 2*a(n) = A093829(12*n + 7).
%F A121361 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - _Amiram Eldar_, Jan 20 2025
%e A121361 G.f. = 1 + x + x^2 + x^3 + x^5 + x^6 + 2*x^7 + x^8 + x^10 + x^11 + ...
%e A121361 G.f. = q^7 + q^19 + q^31 + q^43 + q^67 + q^79 + 2*q^91 + q^103 + ...
%t A121361 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6] EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}]; (* _Michael Somos_, Sep 02 2014 *)
%o A121361 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))};
%Y A121361 Cf. A093766, A093829.
%Y A121361 Cf. A000122, A000700, A010054, A121373.
%K A121361 nonn
%O A121361 0,8
%A A121361 _Michael Somos_, Jul 16 2006