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A056541
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a(n) = 2n*a(n-1) + 1 with a(0)=0.
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2
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0, 1, 5, 31, 249, 2491, 29893, 418503, 6696049, 120528883, 2410577661, 53032708543, 1272785005033, 33092410130859, 926587483664053, 27797624509921591, 889523984317490913, 30243815466794691043
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OFFSET
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0,3
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COMMENTS
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if s(n) is a sequence defined as s(0)=x, s(n) = 2n*s(n-1)+k, n>0, then s(n) = 2^n*n!*x + a(n)*k. - Gary Detlefs, Feb 20 2010
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LINKS
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Harvey P. Dale, Table of n, a(n) for n = 0..403
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FORMULA
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a(n) = floor[(sqrt(e)-1)*2^n*n! ] = A010844(n)-A000165(n).
a(n) = Sum[P(n, k) * 2^k {k=0 to n-1}] - Ross La Haye, Sep 15 2004
a(n) = 2^n*n!*Sum_{k=1..n}{1/(k!*2^k)}, with n>=0. - Paolo P. Lava, Apr 26 2010
Conjecture: a(n) +(-2*n-1)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, May 29 2013
E.g.f.: (exp(x)-1)/(1-2*x) = -12*x/(Q(0)+6*x-3*x^2)/(1-2*x), where Q(k) = 2*(4*k+1)*(32*k^2+16*k+x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
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EXAMPLE
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a(3) = 2*3*a(2)+1 = 6*5+1 = 31.
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MATHEMATICA
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s=0; lst={s}; Do[s+=s++n; AppendTo[lst, Abs[s]], {n, 1, 5!, 2}]; lst [Vladimir Joseph Stephan Orlovsky, Oct 23 2008]
nxt[{n_, a_}]:={n+1, 2a(n+1)+1}; NestList[nxt, {0, 0}, 20][[All, 2]] (* or *) With[{nn=20}, CoefficientList[Series[(Exp[x]-1)/(1-2x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Aug 08 2021 *)
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CROSSREFS
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Cf. A002627.
Sequence in context: A276312 A024451 A046852 * A291885 A126121 A167137
Adjacent sequences: A056538 A056539 A056540 * A056542 A056543 A056544
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KEYWORD
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easy,nonn
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AUTHOR
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Henry Bottomley, Jun 20 2000
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EXTENSIONS
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More terms from James A. Sellers, Jul 04 2000
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STATUS
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approved
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