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Euler transform of powers of 2 [ 2,4,8,16,... ].
20

%I #62 Feb 13 2023 16:45:50

%S 1,2,7,20,59,162,449,1200,3194,8348,21646,55480,141152,356056,892284,

%T 2221208,5497945,13533858,33151571,80826748,196219393,474425518,

%U 1142758067,2742784304,6561052331,15645062126,37194451937,88174252924,208463595471,491585775018

%N Euler transform of powers of 2 [ 2,4,8,16,... ].

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A034899/b034899.txt">Table of n, a(n) for n = 0..3150</a> (first 900 terms from Alois P. Heinz)

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

%H N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 83-89.

%H G. S. Venkatesh and Kurusch Ebrahimi-Fard, <a href="https://arxiv.org/abs/2301.04949">A Formal Power Series Approach to Multiplicative Dynamic Feedback</a>, arXiv:2301.04949 [math.OC], 2023.

%H Thomas Wieder, <a href="/A034899/a034899.txt">Additional comments on this sequence</a>

%F G.f.: 1/Product_{n>0} (1-x^n)^(2^n). - _Thomas Wieder_, Mar 06 2005

%F 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

%F G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - 2*x^k))). - _Ilya Gutkovskiy_, Nov 09 2018

%e From _Geoffrey Critzer_, Mar 07 2012: (Start)

%e Per comment in A102866, a(n) is also the number of multisets of binary words of total length n.

%e a(2) = 7 because the multisets are {a,a}, {b,b}, {a,b}, {aa}, {ab}, {ba}, {bb};

%e 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};

%e where the words within each multiset are separated by commas. (End)

%p series(1/product((1-x^(n))^(2^(n)),n=1..20),x=0,12); (Wieder)

%p # second Maple program:

%p with(numtheory):

%p a:= proc(n) option remember;

%p `if`(n=0, 1, add(add(d*2^d, d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 02 2011

%t 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 *)

%o (PARI) m=50; x='x+O('x^m); Vec(prod(k=1,m,1/(1-x^k)^(2^k))) \\ _G. C. Greubel_, Nov 09 2018

%o (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 ~

%Y Cf. A034691, the Euler transform of 1, 2, 4, 8, 16, 32, 64, ...

%Y Cf. A075729, A102866.

%Y Column k=2 of A144074.

%Y Row sums of A055375 and of A209406.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Thomas Wieder_, Mar 06 2005