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Euler transform of powers of 9.
3

%I #19 Nov 10 2018 05:47:08

%S 1,9,126,1623,20583,254493,3091803,36974025,436377771,5091463423,

%T 58811218362,673298882775,7647050353038,86229872235432,

%U 966019964324004,10757807941399023,119146632352548516,1312935665205028374,14400230629085596621,157253909597473608945

%N Euler transform of powers of 9.

%H Alois P. Heinz, <a href="/A144073/b144073.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)^(9^j).

%F a(n) ~ 9^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(9^(m-1)-1)) = 0.0670436814415340801450018457068097893307906... . - _Vaclav Kotesovec_, Mar 14 2015

%F G.f.: exp(9*Sum_{k>=1} x^k/(k*(1 - 9*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->9^j)(n): seq(a(n), n=0..40);

%t nmax = 20; CoefficientList[Series[Product[1/(1-x^j)^(9^j), {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 14 2015 *)

%Y 9th column of A144074.

%Y Cf. A001019 (powers of 9).

%K nonn

%O 0,2

%A _Alois P. Heinz_, Sep 09 2008