A144072 - OEIS

%I #19 Nov 10 2018 05:47:12

%S 1,8,100,1144,12906,141848,1532276,16290920,170938483,1773107760,

%T 18208004664,185316171472,1871103319988,18756665504080,

%U 186798940872312,1849265718114736,18207140415436701,178355043327697976,1738966407826985884,16881111732250394440

%N Euler transform of powers of 8.

%H Alois P. Heinz, <a href="/A144072/b144072.txt">Table of n, a(n) for n = 0..1000</a>

%H N. J. A. Sloane, <a href="/transforms.txt"> Transforms</a>

%F G.f.: Product_{j>0} 1/(1-x^j)^(8^j).

%F a(n) ~  8^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(8^(m-1)-1)) = 0.0772633520042039151361539536110877247158170... . - _Vaclav Kotesovec_, Mar 14 2015

%F G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - 8*x^k))). - _Ilya Gutkovskiy_, Nov 10 2018

%p 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->8^j)(n): seq(a(n), n=0..40);

%t nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(8^j), {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 14 2015 *)

%Y 8th column of A144074.

%Y Cf. A001018 (powers of 8).

%K nonn

%O 0,2

%A _Alois P. Heinz_, Sep 09 2008