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 A247289 Number of weak peaks in all peakless Motzkin paths of length n. 2
 0, 0, 0, 2, 7, 18, 45, 110, 267, 652, 1602, 3960, 9845, 24594, 61689, 155270, 391962, 991968, 2515964, 6393610, 16275174, 41491776, 105922244, 270734244, 692756227, 1774418286, 4549173861, 11672860634, 29975156134, 77029918152, 198083586300, 509692521982 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A weak peak of a Motzkin path is a vertex on the top of a hump. A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the peakless Motzkin path uhu*h*ddu*h*h*d, where u=(1,1), h=(1,0), d(1,-1), has 5 weak peaks (shown by the stars). a(n) = Sum(k*A247288(n,k), 0<=k<=n-1). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA G.f.: (2-z)*z^3*g/((1-z)^2*(1-z+z^2-2*z^2*g)), where g is defined by g = 1 + z*g + z^2*g*(g-1). D-finite with recurrence n*(n-1)*a(n) +(-7*n^2+28*n-31)*a(n-1) +(n-2)*(13*n-48)*a(n-2) +(-5*n^2+21*n-6)*a(n-3) +(7*n^2-43*n+82)*a(n-4) -(13*n-24)*(n-5)*a(n-5) +(4*n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jul 24 2022 EXAMPLE a(4)=7 because the peakless Motzkin paths u*h*dhh, hu*h*dh, and u*h*h*d have 0, 2, 2, and 3 weak peaks (shown by the stars). MAPLE f := (2-z)*z^3*g/((1-z)^2*(1-z+z^2-2*z^2*g)): eqg := g = 1+z*g+z^2*g*(g-1): g := RootOf(eqg, g): fser := series(f, z = 0, 35): seq(coeff(fser, z, n), n = 0 .. 33); CROSSREFS Cf. A004148, A247288. Sequence in context: A243717 A337482 A174192 * A161870 A362126 A072338 Adjacent sequences: A247286 A247287 A247288 * A247290 A247291 A247292 KEYWORD nonn AUTHOR Emeric Deutsch, Sep 14 2014 STATUS approved

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Last modified June 3 15:36 EDT 2023. Contains 363116 sequences. (Running on oeis4.)