This site is supported by donations to The OEIS Foundation.

 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!)
 A191309 Number of peaks at height >= 2 in all dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights). 2
 0, 0, 0, 0, 1, 2, 8, 16, 47, 94, 244, 488, 1186, 2372, 5536, 11072, 25147, 50294, 112028, 224056, 491870, 983740, 2135440, 4270880, 9188406, 18376812, 39249768, 78499536, 166656772, 333313544, 704069248, 1408138496, 2961699667, 5923399334, 12412521388, 24825042776 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Also number of valleys (i.e., DU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+1+0 = 1 valley. Also number of doublerises (i.e., UU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 = 1 doublerise. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA a(2*n+1) = 2*a(2*n). a(2*n+4) = A029760(n). G.f.: g = 2*z^2*(1-q)/(q*(1-2*z+q)^2), where q=sqrt(1-4*z^2). a(n) ~ 2^(n-3/2)*sqrt(n)/sqrt(Pi) * (1-sqrt(2*Pi/n)). - Vaclav Kotesovec, Mar 20 2014 Conjecture: n*(n-4)*a(n) +(n^2-10*n+15)*a(n-1) +2*(-5*n^2+28*n-27)*a(n-2) -4*(n-3)*(n-8) *a(n-3) +24*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 14 2016 EXAMPLE a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 =1 peak at height >=2. MAPLE q := sqrt(1-4*z^2): g := 2*z^2*(1-q)/(q*(1-2*z+q)^2): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35); MATHEMATICA CoefficientList[Series[2*x^2*(1-Sqrt[1-4*x^2])/(Sqrt[1-4*x^2]*(1-2*x+ Sqrt[1-4*x^2])^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) PROG (PARI) x='x+O('x^50); concat([0, 0, 0, 0], Vec(2*x^2*(1-sqrt(1-4*x^2))/(sqrt(1-4*x^2)*(1-2*x+ sqrt(1-4*x^2))^2))) \\ G. C. Greubel, Mar 26 2017 CROSSREFS Cf. A029760, A191308. Sequence in context: A176143 A296946 A096227 * A323351 A134353 A280229 Adjacent sequences:  A191306 A191307 A191308 * A191310 A191311 A191312 KEYWORD nonn AUTHOR Emeric Deutsch, May 30 2011 STATUS approved

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.

Last modified December 13 09:54 EST 2019. Contains 329968 sequences. (Running on oeis4.)