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A144069 Euler transform of powers of 5.

%I A144069 #28 Sep 08 2022 08:45:38
%S A144069 1,5,40,285,2020,13876,93885,624480,4100470,26609290,170940381,
%T A144069 1088260190,6872684570,43088845030,268374618310,1661512492031,
%U A144069 10229763359245,62663268647185,382039881168240,2318974249801205,14018511922088296,84418983571948025
%N A144069 Euler transform of powers of 5.
%H A144069 Alois P. Heinz, <a href="/A144069/b144069.txt">Table of n, a(n) for n = 0..1000</a>
%H A144069 Vaclav Kotesovec, <a href="https://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
%H A144069 N. J. A. Sloane, <a href="/transforms.txt"> Transforms</a>
%F A144069 G.f.: Product_{j>0} 1/(1-x^j)^(5^j).
%F A144069 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
%F A144069 G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - 5*x^k))). - _Ilya Gutkovskiy_, Nov 09 2018
%p A144069 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);
%t A144069 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_ *)
%t A144069 CoefficientList[Series[Product[1/(1-x^k)^(5^k), {k, 1, 30}], {x, 0, 30}], x] (* _G. C. Greubel_, Nov 09 2018 *)
%o A144069 (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
%o A144069 (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
%Y A144069 5th column of A144074.
%K A144069 nonn
%O A144069 0,2
%A A144069 _Alois P. Heinz_, Sep 09 2008