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A333498 Sum of the heights of all Motzkin paths of length n.

%I A333498 #24 May 10 2020 19:35:18
%S A333498 0,0,1,3,9,25,70,196,552,1560,4423,12573,35826,102310,292786,839554,
%T A333498 2411945,6941593,20011328,57779038,167069317,483739961,1402413161,
%U A333498 4070537585,11827842021,34403798725,100167396088,291903951462,851380987390,2485175809878
%N A333498 Sum of the heights of all Motzkin paths of length n.
%H A333498 Alois P. Heinz, <a href="/A333498/b333498.txt">Table of n, a(n) for n = 0..1000</a>
%H A333498 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%F A333498 a(n) = Sum_{k=1..floor(n/2)} k * A097862(n,k).
%p A333498 b:= proc(x, y, h) option remember; `if`(x=0, h, add(
%p A333498       b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
%p A333498     end:
%p A333498 a:= n-> b(n, 0$2):
%p A333498 seq(a(n), n=0..35);
%t A333498 b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[b[x - 1, y + j, Max[h, y]], {j, -Min[1, y], Min[1, x - y - 1]}]];
%t A333498 a[n_] := b[n, 0, 0];
%t A333498 a /@ Range[0, 35] (* _Jean-François Alcover_, May 10 2020, after Maple *)
%Y A333498 Cf. A001006, A057585, A097862, A136439, A333504, A333608.
%K A333498 nonn
%O A333498 0,4
%A A333498 _Alois P. Heinz_, Mar 24 2020