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A130102
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E.g.f.: (e^x - x)^2.
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5
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1, 0, 2, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072, 8166, 16356, 32738, 65504, 131038, 262108, 524250, 1048536, 2097110, 4194260, 8388562, 16777168, 33554382, 67108812, 134217674, 268435400, 536870854, 1073741764, 2147483586
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of length n binary sequences in which no symbol occurs exactly once. (The Rosenthal formula takes 2^n for the total number of binary sequences and subtracts n for each sequence of length n with a single 0 or 1). - Geoffrey Critzer, Dec 03 2011
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
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FORMULA
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a(n) = 2^n - 2*n for n <> 2 (cf. A005803). - Rainer Rosenthal, Feb 14 2010.
E.g.f.: e^(2*x) - 2*x*e^x + x^2.
a(n) = Sum_{k=0..n} C(n,k)*A060576(k)*A060576(n-k).
G.f. 1 + 2*x^2 - 2*x^3/((2*x - 1)*(x - 1)^2). - R. J. Mathar, Dec 04 2011
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EXAMPLE
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a(4) = 8 because there are 8 sequences on {0,1} such that neither 0 nor 1 occurs exactly once: {0,0,0,0}, {0,0,1,1}, {0,1,0,1}, {0,1,1,0}, {1,0,0,1}, {1,0,1,0}, {1,1,0,0}, {1,1,1,1}. - Geoffrey Critzer, Dec 03 2011
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MATHEMATICA
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a=Exp[x]-x; Range[0, 20]! CoefficientList[Series[a^2, {x, 0, 20}], x] (* Geoffrey Critzer, Dec 03 2011 *)
CoefficientList[Series[1+2*x^2-2*x^3/((2*x-1)*(x-1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 28 2012 *)
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PROG
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(MAGMA) I:=[1, 0, 2, 2, 8, 22]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 28 2012
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CROSSREFS
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Cf. A005803, A060576.
Sequence in context: A208235 A151377 A151407 * A151384 A300460 A290613
Adjacent sequences: A130099 A130100 A130101 * A130103 A130104 A130105
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KEYWORD
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nonn,easy
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AUTHOR
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Paul Barry, May 07 2007
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STATUS
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approved
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