OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
N. J. A. Sloane, Transforms
FORMULA
G.f.: Product_{j>0} 1/(1-x^j)^(5^j).
a(n) ~ 5^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(5^(m-1)-1)) = 0.1412899716579209220312645657307029151422082... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - 5*x^k))). - Ilya Gutkovskiy, Nov 09 2018
MAPLE
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:=n-> etr(j->5^j)(n): seq(a(n), n=0..40);
MATHEMATICA
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a[n_] := etr[Function[5^#]][n]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)
CoefficientList[Series[Product[1/(1-x^k)^(5^k), {k, 1, 30}], {x, 0, 30}], x] (* G. C. Greubel, Nov 09 2018 *)
PROG
(PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^(5^k))) \\ G. C. Greubel, Nov 09 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(5^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 09 2008
STATUS
approved