A144067 Euler transform of powers of 3.
1, 3, 15, 64, 276, 1137, 4648, 18585, 73494, 286834, 1108470, 4243128, 16111333, 60718488, 227302086, 845689753, 3128786415, 11515509603, 42179651417, 153808740042, 558532554942, 2020325112767, 7281212274165, 26151068072301, 93618849857345, 334119804933861
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.
- N. J. A. Sloane, Transforms
Programs
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Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(3^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018 -
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->3^j)(n): seq(a(n), n=0..40);
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Mathematica
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[3^#]][n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *) CoefficientList[Series[Product[1/(1-x^k)^(3^k), {k, 1, 30}], {x, 0, 30}], x] (* G. C. Greubel, Nov 09 2018 *) -
PARI
m=30; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(3^k))) \\ G. C. Greubel, Nov 09 2018
Formula
G.f.: Product_{j>0} 1/(1-x^j)^(3^j).
a(n) ~ 3^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(3^(m-1)-1)) = 0.3047484092142751906436952201501007636114175... . - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - 3*x^k))). - Ilya Gutkovskiy, Nov 09 2018