A034899 Euler transform of powers of 2 [ 2,4,8,16,... ].

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, 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.

Thomas Wieder, Additional comments on this sequence

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); (Wieder)
# second Maple program:
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) m=50; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^(2^k))) \\ G. C. Greubel, Nov 09 2018
(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, ...

Cf. A075729, A102866.

Column k=2 of A144074.

Row sums of A055375 and of A209406.

Sequence in context: A094982 A292400 A007460 * A026124 A026153 A025180

Adjacent sequences: A034896 A034897 A034898 * A034900 A034901 A034902

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Thomas Wieder, Mar 06 2005

Status

approved