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A005803 Second-order Eulerian numbers: a(n) = 2^n - 2*n.
(Formerly M1838)
32
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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 (kitaev(AT)ms.uky.edu), 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

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n=0..500

I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.

Jim Haglund and Mirko Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations.

S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.

S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004). [Broken link]

Sandi Klavžar, Uroš Milutinović and Ciril Petr, Hanoi graphs and some classical numbers, Expo. Math. 23 (2005), no. 4, 371-378.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

FORMULA

G.f.: 1 + 2*x^3/((1-x)^2*(1-2*x)). a(n) = A008517(n-1, 2). - Michael Somos, Oct 13 2002

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

a(n) = A165900(n-1) + Sum_{i=0..n-1} a(i), for n > 0. - Ivan N. Ianakiev, Nov 24 2014

a(n) = A000225(n) - A005408(n-1). - Miquel Cerda, Nov 25 2016

E.g.f.: exp(x)*(exp(x) - 2*x). - Ilya Gutkovskiy, Nov 25 2016

EXAMPLE

G.f. = 1 + 2*x^3 + 8*x^4 + 22*x^5 + 52*x^6 + 114*x^7 + 240*x^8 + 494*x^9 + ...

MAPLE

A005803:=-2*z/(2*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for three leading terms

MATHEMATICA

lst={1, 0}; s=0; Do[s+=(s+=n); AppendTo[lst, s], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)

Table[2^n-2n, {n, 0, 50}] (* or *) LinearRecurrence[{4, -5, 2}, {1, 0, 0}, 51] (* Harvey P. Dale, May 21 2011 *)

PROG

(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'

-- Reinhard Zumkeller, Jan 19 2014

(MAGMA) [2^n-2*n: n in [0..30]]; // Wesley Ivan Hurt, Jun 04 2014

CROSSREFS

Equivalent to second column of A008517.

a(n) = A070313 + 1 = A052515 + 2. Bisection of A077866.

Equals for n =>3 the third right hand column of A163936.

Cf. A000918 (first differences).

Cf. A000079, A005843, A000325.

Sequence in context: A094939 A006732 * A145654 A221880 A074352 A261561

Adjacent sequences:  A005800 A005801 A005802 * A005804 A005805 A005806

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified July 27 04:25 EDT 2017. Contains 289841 sequences.