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Euler transform of powers of 6.
3

%I #21 Nov 10 2018 05:47:20

%S 1,6,57,488,4140,34128,276792,2208312,17389710,135354340,1042965042,

%T 7964675400,60337114778,453795079932,3390657365970,25182770127240,

%U 186007882964211,1366948744701066,9998341947058175,72811720605519840,528078809473488744,3815340122599096360

%N Euler transform of powers of 6.

%H Alois P. Heinz, <a href="/A144070/b144070.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)^(6^j).

%F a(n) ~ 6^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(6^(m-1)-1)) = 0.1108660629759785875628164141261367036457657... . - _Vaclav Kotesovec_, Mar 14 2015

%F G.f.: exp(6*Sum_{k>=1} x^k/(k*(1 - 6*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->6^j)(n): seq(a(n), n=0..40);

%t 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[6^#]][n]; Table[ a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 09 2015, after _Alois P. Heinz_ *)

%Y 6th column of A144074.

%Y Cf. A000400 (powers of 6).

%K nonn

%O 0,2

%A _Alois P. Heinz_, Sep 09 2008