cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A144072 Euler transform of powers of 8.

This page as a plain text file.
%I A144072 #19 Nov 10 2018 05:47:12
%S A144072 1,8,100,1144,12906,141848,1532276,16290920,170938483,1773107760,
%T A144072 18208004664,185316171472,1871103319988,18756665504080,
%U A144072 186798940872312,1849265718114736,18207140415436701,178355043327697976,1738966407826985884,16881111732250394440
%N A144072 Euler transform of powers of 8.
%H A144072 Alois P. Heinz, <a href="/A144072/b144072.txt">Table of n, a(n) for n = 0..1000</a>
%H A144072 N. J. A. Sloane, <a href="/transforms.txt"> Transforms</a>
%F A144072 G.f.: Product_{j>0} 1/(1-x^j)^(8^j).
%F A144072 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 A144072 G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - 8*x^k))). - _Ilya Gutkovskiy_, Nov 10 2018
%p A144072 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 A144072 nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(8^j), {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 14 2015 *)
%Y A144072 8th column of A144074.
%Y A144072 Cf. A001018 (powers of 8).
%K A144072 nonn
%O A144072 0,2
%A A144072 _Alois P. Heinz_, Sep 09 2008