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A113306 Expansion of q * f(-q, -q^11) / f(-q^5, -q^7) in powers of q where f(, ) is Ramanujan's general theta function. 2
1, -1, 0, 0, 0, 1, -1, 1, -1, 0, 1, -2, 2, -2, 2, 0, -2, 3, -4, 4, -2, 0, 2, -5, 7, -6, 3, 0, -4, 8, -10, 9, -6, 0, 8, -12, 14, -14, 9, 0, -10, 18, -22, 20, -12, 0, 13, -26, 33, -29, 17, 0, -20, 37, -45, 42, -26, 0, 29, -52, 62, -58, 37, 0, -40, 72, -88, 80, -48, 0, 53, -99, 122, -110, 65, 0, -72, 134, -163, 148, -91, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,12

COMMENTS

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

Cubic AGM theta functions: a(q) (A004016), b(q) (A005928), c(q) (A005882).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Nicco, A new q-continued fraction of order 12, Mathematics StackExchange, Jul 05 2015.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (c(q) / c(q^4) - phi(q) * psi(q^3) / (q * psi(q^6)^2)) / 2 = 2 / (c(q) / c(q^4) + phi(q) * psi(q^3) / (q * psi(q^6)^2)) in powers of q where c() is a cubic AGM theta function and phi(), psi() are Ramanujan theta functions.

Euler transform of period 12 sequence [-1, 0, 0, 0, 1, 0, 1, 0, 0, 0, -1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + v)^2 - v * (1 + u^2).

G.f.: x * (Product_{k>0} (1 - x^(12*k - 1)) * (1 - x^(12*k - 11)) / ((1 - x^(12*k - 5)) * (1 - x^(12*k - 7)))).

a(2*n) = -A139139(n). a(6*n + 2) = -A139135(n). a(6*n + 4) = 0.

G.f.: x*(1-x) / (1+x^3 - x^3*(1+x^2)*(1+x^4) / (1+x^9 + x^6*(1-x^5)*(1-x^7) / (1+x^15 - ...))) [Nicco 2015]. - Michael Somos, Mar 20 2018

G.f.: x*(1-x) / (1-x^3 + x^3*(1-x^2)*(1-x^4) / ((1-x^3)*(1+x^6) + x^3*(1-x^8)*(1-x^10) / ((1-x^3)*(1+x^12) + ...))) [Piezas 2015]. - Michael Somos, Mar 20 2018

EXAMPLE

G.f. = q - q^2 + q^6 - q^7 + q^8 - q^9 + q^11 - 2*q^12 + 2*q^13 - 2*q^14 + 2*q^15 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ q Product[ (1 - q^k)^KroneckerSymbol[12, k], {k, n - 1}], {q, 0, n}]; (* Michael Somos, Mar 20 2018 *)

a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^12] QPochhammer[ q^11, q^12] / (QPochhammer[ q^5, q^12] QPochhammer[ q^7, q^12]), {q, 0, n}]; (* Michael Somos, Mar 20 2018 *)

PROG

(PARI) {a(n) = if( n<1, 0, n--; polcoeff( prod( k=1, n, (1 - x^k)^kronecker(12, k), 1 + x * O(x^n)), n))};

CROSSREFS

Cf. A139135, A139139.

Sequence in context: A305029 A097033 A268686 * A181089 A341894 A171932

Adjacent sequences:  A113303 A113304 A113305 * A113307 A113308 A113309

KEYWORD

sign

AUTHOR

Michael Somos, Oct 24 2005

STATUS

approved

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Last modified May 18 01:24 EDT 2021. Contains 343992 sequences. (Running on oeis4.)