login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A057585 Area under Motzkin excursions. 2

%I

%S 0,1,4,16,56,190,624,2014,6412,20219,63284,196938,610052,1882717,

%T 5792528,17776102,54433100,166374109,507710420,1547195902,4709218604,

%U 14318240578,43493134160,132003957436,400337992056,1213314272395

%N Area under Motzkin excursions.

%C a(n) is the sum of areas under all Motzkin excursions of length n (nonnegative walks beginning and ending in 0, with jumps -1,0,+1).

%H T. D. Noe, <a href="/A057585/b057585.txt">Table of n, a(n) for n = 1..400</a>

%H C. Banderier, <a href="http://algo.inria.fr/banderier/">Analytic combinatorics of random walks and planar maps</a>, PhD Thesis, 2001.

%F G.f.: (x^2 + 2*x - 1 + (-x+1)*sqrt((x+1)*(1-3*x)))/(2*(3*x-1)*(x+1)*x^2).

%F Recurrence: (n-2)*(n+2)*a(n) = (n+1)*(4*n-7)*a(n-1) + (2*n^2 - 3*n - 8)*a(n-2) - 3*(n-1)*(4*n-5)*a(n-3) - 9*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Sep 11 2013

%F a(n) ~ 3^(n+1)/4 * (1-2*sqrt(3)/sqrt(Pi*n)). - _Vaclav Kotesovec_, Sep 11 2013

%p G:= (x^2+2*x-1+(-x+1)*sqrt((x+1)*(1-3*x)))/(2*(3*x-1)*(x+1)*x^2): Gser:=series(G,x=0,30): seq(coeff(Gser,x,n),n=1..26); # _Emeric Deutsch_, Apr 08 2007

%t f[x_] := (x^2+2*x-1+(-x+1)*Sqrt[(x+1)*(1-3*x)]) / (2*(3*x-1)*(x+1)*x^2); Drop[ CoefficientList[ Series[ f[x], {x, 0, 26}], x], 1] (* _Jean-Fran├žois Alcover_, Dec 21 2011, from g.f. *)

%K easy,nonn,nice

%O 1,3

%A _Cyril Banderier_, Oct 04 2000

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 18:49 EST 2019. Contains 329768 sequences. (Running on oeis4.)