A144068 Euler transform of powers of 4.

1, 4, 26, 148, 843, 4632, 25124, 133784, 703553, 3655340, 18800886, 95819580, 484416675, 2431094352, 12120072472, 60058765072, 295959923287, 1450980481036, 7079894939166, 34393241899772, 166390593502701, 801877654792696, 3850469199935412, 18426281811165880

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)^(4^j).

a(n) ~ 4^n * exp(2*sqrt(n) - 1/2 + c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} 1/(m*(4^(m-1)-1)) = 0.1938490811676466793200632998157568919969827... . - Vaclav Kotesovec, Mar 14 2015

G.f.: exp(4*Sum_{k>=1} x^k/(k*(1 - 4*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->4^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[4^#]][n]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)
CoefficientList[Series[Product[1/(1-x^k)^(4^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)^(4^k))) \\ G. C. Greubel, Nov 09 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(4^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

Crossrefs

4th column of A144074.

Sequence in context: A180226 A325587 A223627 * A204062 A121767 A092167

Adjacent sequences: A144065 A144066 A144067 * A144069 A144070 A144071

Keyword

nonn

Author

Alois P. Heinz, Sep 09 2008

Status

approved