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URL: https://oeis.org/A229793

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A229793
The expansion of R(q)^-5 in powers of q where R() is the Rogers-Ramanujan continued fraction.
7
1, 5, 10, 5, -15, -24, 15, 70, 30, -125, -175, 95, 420, 180, -615, -826, 410, 1760, 705, -2415, -3100, 1530, 6270, 2460, -8090, -10174, 4840, 19570, 7500, -24360, -30024, 14130, 55970, 21155, -67380, -81926, 37895, 148410, 55305, -174500, -209577, 96025
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 8 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - Michael Somos, Jul 22 2014
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(5). [Yang 2004] - Michael Somos, Jul 22 2014
LINKS
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
FORMULA
Expansion of (1/q) * (f(-q^2, -q^3) / f(-q, -q^4))^5 in powers of q where f() is a Ramanujan theta function.
Euler transform of period 5 sequence [ 5, -5, -5, 5, 0, ...].
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = u^2 + v - u*v^2 * (u^2 - v) + 10 * u*v * (1 + u - v + u*v).
G.f.: (1/x) * exp( integrate_{0..x} (1 - eta(x)^5 / eta(x^5)) dx / x ).
G.f.: (1/x) * (Product_{k>0} (1 - x^{5*k - 2}) * (1 - x^{5*k - 3}) / (1 - x^{5*k - 1}) * (1 - x^{5*k - 4}))^5.
G.f.: (1/x) * ( (Sum_{k in Z} (-1)^n * x^((5*k + 1) * k/2)) / (Sum_{k in Z} (-1)^n * x^((5*k + 3) * k/2)))^5.
Convolution inverse of A078905.
a(-1) = 1, a(n) = (5/(n+1))*Sum_{k=1..n+1} A109091(k)*a(n-k) for n > -1. - Seiichi Manyama, Apr 01 2017
EXAMPLE
G.f. = 1/q + 5 + 10*q + 5*q^2 - 15*q^3 - 24*q^4 + 15*q^5 + 70*q^6 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5] / QPochhammer[ q, q^5] QPochhammer[ q^4, q^5])^5, {q, 0, n}];
PROG
(PARI) {a(n) = local(A); if( n<-1, 0, A = x^2 * O(x^n); A = (eta(x + A) / eta(x^5 + A))^6 / x; polcoeff( (11 + A + sqrt(125 + 22*A + A^2)) / 2, n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 29 2013
STATUS
approved