%I #12 Jul 24 2022 12:43:13
%S 0,0,0,2,7,18,45,110,267,652,1602,3960,9845,24594,61689,155270,391962,
%T 991968,2515964,6393610,16275174,41491776,105922244,270734244,
%U 692756227,1774418286,4549173861,11672860634,29975156134,77029918152,198083586300,509692521982
%N Number of weak peaks in all peakless Motzkin paths of length n.
%C A weak peak of a Motzkin path is a vertex on the top of a hump.
%C 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).
%C a(n) = Sum(k*A247288(n,k), 0<=k<=n-1).
%H Alois P. Heinz, <a href="/A247289/b247289.txt">Table of n, a(n) for n = 0..1000</a>
%F 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).
%F 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
%e 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).
%p 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);
%Y Cf. A004148, A247288.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Sep 14 2014