A127692 Expansion of psi(x^4) + x * psi(x^12) in powers of x where psi() is a Ramanujan theta function.

1, 1, 0, 0, 1, 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, 1, 0, 0, 1, 0, 0, 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, 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, 0, 0, 0

Offset

0,1

Comments

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

a(n) = 1 if n is four times a triangular number or one more than twelve times a triangular number else 0. - Michael Somos, Jul 19 2012

Links

Antti Karttunen, Table of n, a(n) for n = 0..10000

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 5.

Michael Somos, Introduction to Ramanujan theta functions, 2019.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Euler transform of period 24 sequence [ 1, -1, 0, 1, -1, 1, -1, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 1, -1, 1, 0, -1, 1, -1, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, else b(p^e) = (1 + (-1)^e)/2.

a(3*n + 1) = a(n), a(3*n + 2) = a(4*n + 2) = a(4*n + 3) = a(6*n + 3) = 0.

a(2*n) = A005369(n). a(4*n) = A010054(n). a(6*n) = A089806(n). a(12*n) = A080995(n).

G.f.: Sum_{k>0} x^(2k(k-1)) +x^(6k(k-1)+1) = Product_{k>0} (1-x^(24k)) (1-x^(24k-5)) (1-x^(24k-7)) (1-x^(24k-17)) (1-x^(24k-19)) (1+x^(12k-1)) (1+x^(12k-4)) (1+x^(12k-6)) (1+x^(12k-8)) (1+x^(12k-11)).

From Michael Somos, Jul 19 2012: (Start)

Expansion of f(x, -x^5) * f(-x^4, -x^8) / f(x, -x) in powers of x where f(,) is the Ramanujan two-variable theta function.

G.f.: (Sum_{k in Z} x^(2*k*(k + 1)) + x^(6*k*(k + 1) + 1)) / 2.

a(n) = A195198(2*n + 1). (End)

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1/sqrt(2) + 1/sqrt(6) = 1.115355... (A145439). - Amiram Eldar, Dec 29 2023

Example

1 + x + x^4 + x^12 + x^13 + x^24 + x^37 + x^40 + x^60 + x^73 + x^84 + ...
q + q^3 + q^9 + q^25 + q^27 + q^49 + q^75 + q^81 + q^121 + q^147 + q^169 + ...

Prog

(PARI) {a(n) = issquare(2*n + 1) + issquare(6*n + 3)}
(PARI) {a(n) = n = 2*n + 1; issquare(n) || issquare(3*n)}

Crossrefs

Cf. A005369, A010054, A080995, A089806, A145439.

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A138149 A113047 A214505 * A014305 A023533 A010052

Adjacent sequences: A127689 A127690 A127691 * A127693 A127694 A127695

Keyword

nonn,easy

Author

Michael Somos, Jan 19 2007

Status

approved