A127693 Expansion of psi(x^2) + x * psi(x^10) in powers of x where psi() is a Ramanujan theta function.

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, 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, 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

Offset: 0

Author

Michael Somos, Jan 19 2007

Keywords

• nonn
• easy

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + x^2 + x^6 + x^11 + x^12 + x^20 + x^30 + x^31 + x^42 + x^56 + x^61 + ...
G.f. = q + q^5 + q^9 + q^25 + q^45 + q^49 + q^81 + q^121 + q^125 + q^169 + q^225 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Richard Blecksmith, John Brillhart, and Irving Gerst,, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)
• Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001), 131-139. see p. 134 Theorem 6.
• Michael Somos, Introduction to Ramanujan theta functions, 2019.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Crossrefs

Cf. A010054, A080995, A089806, A344212.

Cf. A000122, A000700, A121373.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] + EllipticTheta[ 2, 0, x^5]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Jul 08 2015 *)

PARI

{a(n) = issquare(4*n + 1) + issquare(20*n + 5)};

Formula

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

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, ...].

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.

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.

G.f.: Sum_{k>0} x^(k*(k - 1)) + x^(5*k*(k - 1) + 1).

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)).

a(2*n) = A010054(n). a(3*n) = A089806(n). a(6*n) = A080995(n).

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1 + 1/sqrt(5) = 1.447213... (A344212). - Amiram Eldar, Dec 29 2023