A144069 Euler transform of powers of 5.

1, 5, 40, 285, 2020, 13876, 93885, 624480, 4100470, 26609290, 170940381, 1088260190, 6872684570, 43088845030, 268374618310, 1661512492031, 10229763359245, 62663268647185, 382039881168240, 2318974249801205, 14018511922088296, 84418983571948025

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

5th column of A144074.

Sequence in context: A124306 A124545 A125729 * A280158 A073505 A145841

Adjacent sequences: A144066 A144067 A144068 * A144070 A144071 A144072

Keyword

nonn

Author

Alois P. Heinz, Sep 09 2008

Status

approved