Search: id:A005803
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%I A005803 M1838
%S A005803 1,0,0,2,8,22,52,114,240,494,1004,2026,4072,8166,16356,32738,65504,
%T A005803 131038,262108,524250,1048536,2097110,4194260,8388562,16777168,
%U A005803 33554382,67108812,134217674,268435400,536870854,1073741764,2147483586
%N A005803 Second-order Eulerian numbers: 2^n - 2n.
%C A005803 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 (kitaev(AT)ms.uky.edu), Nov 11 2004
%C A005803 No multiplications, no exponentiation is required! Simplify! [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008]
%D A005803 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005803 I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, 
               A 24 (1978), 24-33.
%D A005803 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 256.
%D A005803 Klavzar, S.; Milutinovic, U.; and Petr, C., Hanoi graphs and some classical 
               numbers, Expo. Math. 23 (2005), no. 4, 371-378.
%H A005803 T. D. Noe, <a href="b005803.txt">Table of n, a(n) for n=0..500</a>
%H A005803 S. Kitaev, <a href="http://www.integers-ejcnt.org/vol4.html">On multi-avoidance 
               of right angled numbered polyomino patterns</a>, Integers: Electronic 
               Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
%H A005803 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005803 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005803 S. Kitaev, <a href="http://www.ms.uky.edu/%7Emath/MAreport/4-ser.ps">
               On multi-avoidance of right angled numbered polyomino patterns</a>
               , University of Kentucky Research Reports (2004).
%F A005803 G.f.: 1 + 2x^3/((1-x)^2(1-2x)). a(n)=A008517(n-1, 2). - Michael Somos, 
               Oct 13, 2002
%F A005803 Equals binomial transform of [1, -1, 1, 1, 1,...]. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jul 14 2008
%F A005803 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%F A005803 a(0) = 1 and a(n) = sum((-1)^(n+k+1)*binomial(2*n-1,k)*stirling1(2*n-k-3,
               n-k-2),k=0..n-3), n=>1.
%F A005803 (End)
%p A005803 a:=n->sum (2^j-2,j=2..n): seq(a(n),n=-1..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 27 2007
%p A005803 A005803:=-2*z/(2*z-1)/(z-1)**2; [Conjectured by S. Plouffe in his 1992 
               dissertation. Gives sequence except for three leading terms.]
%t A005803 lst={1, 0};s=0;Do[s+=(s+=n);AppendTo[lst, s], {n, 0, 5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008]
%o A005803 (PARI) a(n)=if(n<0,0,2^n-2*n)
%Y A005803 a(n) = A070313 + 1 = A052515 + 2. Bisection of A077866.
%Y A005803 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 
               2009: (Start)
%Y A005803 Equals for n =>3 the third right hand column of A163936.
%Y A005803 (End)
%Y A005803 Sequence in context: A006696 A094939 A006732 this_sequence A145654 A074352 
               A017928
%Y A005803 Adjacent sequences: A005800 A005801 A005802 this_sequence A005804 A005805 
               A005806
%K A005803 nonn,easy,nice
%O A005803 0,4
%A A005803 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

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