OFFSET
1,12
COMMENTS
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)))).
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
KEYWORD
sign
AUTHOR
Michael Somos, Oct 24 2005
STATUS
approved