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A005803
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Second-order Eulerian numbers: a(n) = 2^n - 2*n.
(Formerly M1838)
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39
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1, 0, 0, 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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0,4
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COMMENTS
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Starting with n=2, a(n) is the second-order Eulerian number <<n-1,1>> (see A008517).
Also, number of 3 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (01;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
This is the number of target DNA sequences of the correct length present after each successive cycle of the Polymerase Chain Reaction (PCR). The first two cycles produce 0 target sequences, there are 2 target sequences present after the third cycle, then 8 after the fourth cycle, and so forth. - A. Timothy Royappa, Jun 16 2012
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Equals binomial transform of [1, -1, 1, 1, 1, ...]. - Gary W. Adamson, Jul 14 2008
a(0) = 1 and a(n) = Sum_{k=0..n-3} ((-1)^(n+k+1)*binomial(2*n-1,k)*stirling1(2*n-k-3,n-k-2)), n=>1. - Johannes W. Meijer, Oct 16 2009
a(0)=1, a(1)=0, a(2)=0, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, May 21 2011
a(n) = A000225(n+1) - A081494(n+1), n > 1. In other words, a(n) equals the sum of the elements in a Pascal triangle of depth n+1 minus the sum of the elements on its perimeter. - Ivan N. Ianakiev, Jun 01 2014
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EXAMPLE
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G.f. = 1 + 2*x^3 + 8*x^4 + 22*x^5 + 52*x^6 + 114*x^7 + 240*x^8 + 494*x^9 + ...
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MAPLE
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A005803:=-2*z/(2*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for three leading terms
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MATHEMATICA
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Table[2^n-2n, {n, 0, 50}] (* or *) LinearRecurrence[{4, -5, 2}, {1, 0, 0}, 51] (* Harvey P. Dale, May 21 2011 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, 2^n - 2*n)}; /* Michael Somos, Oct 13 2002 */
(Haskell)
a005803 n = 2 ^ n - 2 * n
a005803_list = 1 : f 1 [0, 2 ..] where
f x (z:zs@(z':_)) = y : f y zs where y = (x + z) * 2 - z'
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CROSSREFS
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Equivalent to second column of A008517.
Equals for n =>3 the third right hand column of A163936.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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