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

0,1

Comments

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

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.

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A010054, A080995, A089806, A344212.

Cf. A000122, A000700, A121373.

Sequence in context: A016415 A266659 A214960 * A129405 A127001 A266848

Adjacent sequences: A127690 A127691 A127692 * A127694 A127695 A127696

Keyword

nonn,easy

Author

Michael Somos, Jan 19 2007

Status

approved