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A144068 Euler transform of powers of 4. 3
1, 4, 26, 148, 843, 4632, 25124, 133784, 703553, 3655340, 18800886, 95819580, 484416675, 2431094352, 12120072472, 60058765072, 295959923287, 1450980481036, 7079894939166, 34393241899772, 166390593502701, 801877654792696, 3850469199935412, 18426281811165880 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)