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

0,2

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

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

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);

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 *)

Prog

(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
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(3^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

Crossrefs

3rd column of A144074. Row sums of A275414.

Cf. A256142.

Sequence in context: A151241 A080948 A098102 * A001447 A106732 A052981

Adjacent sequences: A144064 A144065 A144066 * A144068 A144069 A144070

Keyword

nonn

Author

Alois P. Heinz, Sep 09 2008

Status

approved