OFFSET
0,15
COMMENTS
REFERENCES
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162.
LINKS
Michael Somos, Introduction to Ramanujan theta functions [This is just section 1 of the next item, but it is mentioned in over 1000 sequences. - N. J. A. Sloane, Nov 13 2019]
Michael Somos, A Multisection of q-Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Euler transform of period 7 sequence [ 1, -1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Oct 03 2013
Given g.f. A(x), then B(q) = q^-2*A(q^7) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^3 - u^6 + 3*u^4*v + u^7*v^3 + u^2*v^9 + u^8*v^6 - 3*u^2*v^2 - 2*u*v^6 - 5*u^3*v^5 - u^5*v^4 - u^9*v^2 - u^4*v^8 - u^6*v^7.
G.f.: Product_{k>0} (1 - x^(7*k - 2)) * (1 - x^(7*k - 5)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6))).
a(n) = A229894(7*n). - Michael Somos, Oct 03 2013
G.f.: B(x) / C(x), where B(x) is the g.f. of A375106 and C(x) is the g.f. of A375150. - Seiichi Manyama, Aug 03 2024
EXAMPLE
G.f. = 1 + x - x^5 + x^7 + x^8 - x^10 - x^11 - x^12 + 2*x^14 + 2*x^15 + ...
G.f. = q^-2 + q^5 - q^33 + q^47 + q^54 - q^68 - q^75 - q^82 + 2*q^96 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] / (QPochhammer[ x, x^7] QPochhammer[ x^6, x^7]), {x, 0, n}]; (* Michael Somos, Oct 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 0, 0, 1, -1, 0}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 03 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[ 0, -1, 1, 0, 0, 1, -1][k%7 + 1]), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jun 04 2005
STATUS
approved