OFFSET
-1,41
REFERENCES
G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005, see p. 11, Equation (1.1.10)
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
T. Horie and N. Kanou, Certain modular functions similar to the Dedekind eta function, Abh. Math. Sem. Univ. Hamburg 72 (2002), 89-117. MR1941549 (2003j:11043)
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, see p. 238, Equation (20.2)
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q) / eta(q^25) = (1/q) * f(-q) / f(-q^25) in powers of q where f() is a Ramanujan theta function.
Euler transform of period 25 sequence [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, ...].
G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 - v) * (u - v^2) - 2*u*v * (u + v + 2).
G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^4)) where f(u, v, w) = u^2 + u*w + w^2 - v*(2*(u + w) + 5) - v^2*(u + w + 2).
G.f.: x^-1 * Product_{k>0} (1 - x^k) / (1 - x^(25*k)).
Expansion of 1/R(q) - 1 - R(q) in powers of q where R() is the g.f. of A007325 the Rogers-Ramanujan continued fraction. - Michael Somos, May 09 2016
a(-1) = 1, a(n) = -(1/(n+1))*Sum_{k=1..n+1} A227131(k)*a(n-k) for n > -1. - Seiichi Manyama, Mar 29 2017
EXAMPLE
G.f. = 1/q - 1 - q + q^4 + q^6 - q^11 - q^14 + q^21 + q^24 - q^26 + q^29 + ...
MATHEMATICA
a[ n_] := With[ {m = n + 1}, SeriesCoefficient[ Product[ 1 - q^k, {k, m}] / Product[ 1 - q^k, {k, 25, m, 25}], {q, 0, m}]];
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q] / QPochhammer[ q^25]), {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)
PROG
(PARI) {a(n) = my(A, m); if( n<-1, 0, m=5; A = x + O(x^6); while( m < n + 2, m*=5; A = x * subst((A * (1 - 2*A + 4*A^2 - 3*A^3 + A^4) / (1 + 3*A+ 4*A^2 + 2*A^3 + A^4) / x)^(1/5), x, x^5)); polcoeff( 1/A - A - 1, n))};
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) / eta(x^25 + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 02 2004
STATUS
approved