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A034899
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Euler transform of powers of 2 [ 2,4,8,16,... ].
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20
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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 ~
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CROSSREFS
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Cf. A034691, the Euler transform of 1, 2, 4, 8, 16, 32, 64, ...
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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