A144072 Euler transform of powers of 8.
1, 8, 100, 1144, 12906, 141848, 1532276, 16290920, 170938483, 1773107760, 18208004664, 185316171472, 1871103319988, 18756665504080, 186798940872312, 1849265718114736, 18207140415436701, 178355043327697976, 1738966407826985884, 16881111732250394440
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- N. J. A. Sloane, Transforms
Programs
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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->8^j)(n): seq(a(n), n=0..40);
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Mathematica
nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(8^j), {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 14 2015 *)
Formula
G.f.: Product_{j>0} 1/(1-x^j)^(8^j).
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
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - 8*x^k))). - Ilya Gutkovskiy, Nov 10 2018