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URL: https://oeis.org/A034899

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A034899
Euler transform of powers of 2 [ 2,4,8,16,... ].
20
1, 2, 7, 20, 59, 162, 449, 1200, 3194, 8348, 21646, 55480, 141152, 356056, 892284, 2221208, 5497945, 13533858, 33151571, 80826748, 196219393, 474425518, 1142758067, 2742784304, 6561052331, 15645062126, 37194451937, 88174252924, 208463595471, 491585775018
OFFSET
0,2
LINKS
Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..3150 (first 900 terms from Alois P. Heinz)
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 and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
G. S. Venkatesh and Kurusch Ebrahimi-Fard, A Formal Power Series Approach to Multiplicative Dynamic Feedback, arXiv:2301.04949 [math.OC], 2023.
FORMULA
G.f.: 1/Product_{n>0} (1-x^n)^(2^n). - Thomas Wieder, Mar 06 2005
a(n) ~ c^2 * 2^(n-1) * exp(2*sqrt(n) - 1/2) / (sqrt(Pi) * n^(3/4)), where c = A247003 = exp( Sum_{k>=2} 1/(k*(2^k-2)) ) = 1.3976490050836502... . - Vaclav Kotesovec, Mar 09 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - 2*x^k))). - Ilya Gutkovskiy, Nov 09 2018
EXAMPLE
From Geoffrey Critzer, Mar 07 2012: (Start)
Per comment in A102866, a(n) is also the number of multisets of binary words of total length n.
a(2) = 7 because the multisets are {a,a}, {b,b}, {a,b}, {aa}, {ab}, {ba}, {bb};
a(3) = 20 because the multisets are {a,a,a}, {b,b,b}, {a,a,b}, {a,b,b}, {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb};
where the words within each multiset are separated by commas. (End)
MAPLE
series(1/product((1-x^(n))^(2^(n)), n=1..20), x=0, 12); # Thomas Wieder
# Alternative:
with(numtheory):
a:= proc(n) option remember;
`if`(n=0, 1, add(add(d*2^d, d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Sep 02 2011
MATHEMATICA
nn = 20; p = Product[1/(1 - x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)
PROG
(PARI) my(m=50, x='x+O('x^m)); Vec(prod(k=1, m, 1/(1-x^k)^(2^k))) \\ G. C. Greubel, Nov 09 2018
(PARI) ET(v) = Vec(prod(k=1, #v, 1/(1-x^k+x*O(x^#v))^v[k]))
aList(upto) = ET([2^n | n <- [1..upto]]) \\ Peter Luschny, Jan 08 2026
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[1/(1-x^k)^(2^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018 ~
CROSSREFS
Cf. A034691, the Euler transform of 1, 2, 4, 8, 16, 32, 64, ...
Column k=2 of A144074.
Row sums of A055375 and of A209406.
Sequence in context: A292400 A389120 A007460 * A026124 A026153 A025180
KEYWORD
nonn
EXTENSIONS
More terms from Thomas Wieder, Mar 06 2005
STATUS
approved