%I #25 Dec 29 2023 03:50:24
%S 1,1,1,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,
%T 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,
%U 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0
%N Expansion of psi(x^2) + x * psi(x^10) in powers of x where psi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A127693/b127693.txt">Table of n, a(n) for n = 0..1000</a>
%H Richard Blecksmith, John Brillhart, and Irving Gerst,, <a href="http://dx.doi.org/10.1090/S0025-5718-1988-0942157-2">Some infinite product identities</a>, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)
%H Shaun Cooper and Michael Hirschhorn, <a href="http://dx.doi.org/10.1216/rmjm/1008959672">On some infinite product identities</a>, Rocky Mountain J. Math., 31 (2001), 131-139. see p. 134 Theorem 6.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of f(-x^2, -x^3) * f(x, -x^4) / f(-x^2, -x^2) = f(x^2, -x^3) * f(x, x^4) / f(-x^10, -x^10) where f(,) is Ramanujan's general theta function. - _Michael Somos_, Jul 30 2012
%F Euler transform of period 20 sequence [ 1, 0, -1, 0, 0, 1, -1, 0, 1, -1, 1, 0, -1, 1, 0, 0, -1, 0, 1, -1, ...].
%F a(n) = b(4*n + 1) where b() is multiplicative and b(2^e) = 0^e, b(5^e) = 1, else b(p^e) = (1 + (-1)^e) / 2.
%F a(9*n + 2) = a(5*n + 1) = a(n), a(5*n + 3) = a(5*n + 4) = a(6*n + 3) = a(6*n + 4) = a(9*n + 5) = a(9*n + 8) = 0.
%F G.f.: Sum_{k>0} x^(k*(k - 1)) + x^(5*k*(k - 1) + 1).
%F G.f.: Product_{k>0} (1 - x^(10*k)) * (1 + x^(10*k - 1)) * (1 + x^(10*k-2)) * (1 - x^(10*k - 3)) * (1 + x^(10*k - 4)) * (1 + x^(10*k - 6)) * (1 - x^(10*k - 7)) * (1 + x^(10*k -8)) * (1 + x^(10*k - 9)).
%F a(2*n) = A010054(n). a(3*n) = A089806(n). a(6*n) = A080995(n).
%F Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1 + 1/sqrt(5) = 1.447213... (A344212). - _Amiram Eldar_, Dec 29 2023
%e G.f. = 1 + x + x^2 + x^6 + x^11 + x^12 + x^20 + x^30 + x^31 + x^42 + x^56 + x^61 + ...
%e G.f. = q + q^5 + q^9 + q^25 + q^45 + q^49 + q^81 + q^121 + q^125 + q^169 + q^225 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] + EllipticTheta[ 2, 0, x^5]) / (2 x^(1/4)), {x, 0, n}]; (* _Michael Somos_, Jul 08 2015 *)
%o (PARI) {a(n) = issquare(4*n + 1) + issquare(20*n + 5)};
%Y Cf. A010054, A080995, A089806, A344212.
%Y Cf. A000122, A000700, A121373.
%K nonn,easy
%O 0,1
%A _Michael Somos_, Jan 19 2007