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A130102 E.g.f.: (e^x - x)^2. 4

%I

%S 1,0,2,2,8,22,52,114,240,494,1004,2026,4072,8166,16356,32738,65504,

%T 131038,262108,524250,1048536,2097110,4194260,8388562,16777168,

%U 33554382,67108812,134217674,268435400,536870854,1073741764,2147483586

%N E.g.f.: (e^x - x)^2.

%C 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

%H Vincenzo Librandi, <a href="/A130102/b130102.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).

%F a(n) = 2^n - 2*n for n <> 2 (cf. A005803). - _Rainer Rosenthal_, Feb 14 2010.

%F E.g.f.: e^(2*x) - 2*x*e^x + x^2.

%F a(n) = Sum_{k=0..n} C(n,k)*A060576(k)*A060576(n-k).

%F G.f. 1 + 2*x^2 - 2*x^3/((2*x - 1)*(x - 1)^2). - _R. J. Mathar_, Dec 04 2011

%e 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

%t a=Exp[x]-x; Range[0,20]! CoefficientList[Series[a^2, {x,0,20}], x] (* _Geoffrey Critzer_, Dec 03 2011 *)

%t CoefficientList[Series[1+2*x^2-2*x^3/((2*x-1)*(x-1)^2),{x,0,40}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)

%o (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

%Y Cf. A005803, A060576.

%K nonn,easy

%O 0,3

%A _Paul Barry_, May 07 2007

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Last modified May 22 15:27 EDT 2018. Contains 304426 sequences. (Running on oeis4.)