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A127693 Expansion of psi(x^2) + x * psi(x^10) in powers of x where psi() is a Ramanujan theta function. 3
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 (list; graph; refs; listen; history; text; internal format)
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

R. Blecksmith; J. Brillhart; I. Gerst, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301-314. MR0942157 (89f:05017)

S. Cooper and M. 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

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).

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.

Sequence in context: A016415 A266659 A214960 * A129405 A127001 A266848

Adjacent sequences: A127690 A127691 A127692 * A127694 A127695 A127696

KEYWORD

nonn

AUTHOR

Michael Somos, Jan 19 2007

STATUS

approved

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Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)