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A120765
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Expansion of e.g.f. -exp(-x)*log(1-2*x)/2.
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3
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0, 1, 0, 5, 24, 209, 2120, 25829, 365456, 5895105, 106794992, 2147006949, 47436635752, 1142570789073, 29797622256376, 836527783016197, 25153234375160992, 806519154686509057, 27470342073410272608, 990496662138073867333, 37692249497898323450424
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OFFSET
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0,4
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COMMENTS
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Inverse binomial transform of even double factorials (A000165) with 0 prepended: [0, 1, 2, 8, 48 ...].
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LINKS
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FORMULA
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E.g.f.: -e^(-x)*log(1-2*x)/2.
a(n) = Sum_{i=0..n-1} (-1)^(n-1-i) * C(n,i+1) * i! * 2^i.
a(n) = Sum_{k=0..n-1} A000354(k)*(-1)^(n+k+1).
Recurrence: a(0) = 0, a(1) = 1, a(2) = 0, a(n) = 2*(n-2)*a(n-3) + (4*n-7)*a(n-2) + 2*(n-2)*a(n-1). (End)
a(n) = (-1)^(n+1)*n*hypergeom([1-n,1,1],[2],2). - Peter Luschny, May 09 2017
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MAPLE
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a:= proc(n) option remember; `if`(n<3, n*(2-n),
(2*n-4)*(a(n-1) +a(n-3)) +(4*n-7)*a(n-2))
end:
A120765 := n -> (-1)^(n+1)*n*hypergeom([1-n, 1, 1], [2], 2):
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MATHEMATICA
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CoefficientList[Series[-E^(-x)*Log[1-2*x]/2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)
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PROG
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(PARI) x='x+O('x^33); concat([0], Vec(serlaplace(-exp(-x)*log(1-2*x)/2))) \\ Joerg Arndt, Jun 29 2015
(PARI) vector(30, n, n--; sum(k=0, n-1, (-1)^(n-1-k) * binomial(n, k+1) * k! * 2^k)) \\ Altug Alkan, Oct 28 2015
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CROSSREFS
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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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