OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
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).
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
FORMULA
Expansion of q^(-1) * eta(q^25) / eta(q) in powers of q.
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, ...].
Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - 5*(u*v)^2 - 2*u*v *(u+v) - u*v.
G.f.: Product_{k>0} (1 - x^(25*k)) / (1 - x^k).
a(n) ~ exp(4*Pi*sqrt(n)/5) / (5*sqrt(10)*n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
a(n) = (1/n)*Sum_{k=1..n} A227131(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Jun 16 2017
EXAMPLE
G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + ...
G.f. = q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 7*q^6 + 11*q^7 + 15*q^8 + 22*q^9 + 30*q^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 25, n, 25}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[(QPochhammer[ x^25] / QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 13 2014 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^25 + A) / eta(x + A), n))};
(PARI) {a(n) = local(A, m); if( n<0, 0, n++; m=5; A = x + O(x^6); while( m<n, 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 / (1/A - A -1), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 10 2004
STATUS
approved