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 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

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Last modified May 28 23:06 EDT 2023. Contains 363028 sequences. (Running on oeis4.)