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A372033 The total number of levels visited by all Motzkin paths of length n.

%I A372033 #13 Apr 16 2024 19:05:17
%S A372033 1,1,3,7,18,46,121,323,875,2395,6611,18371,51337,144145,406420,
%T A372033 1150126,3265412,9298372,26547710,75978322,217921336,626287520,
%U A372033 1803176384,5200298000,15020569818,43447201226,125837214564,364911724264,1059404265599,3078918594707
%N A372033 The total number of levels visited by all Motzkin paths of length n.
%H A372033 Alois P. Heinz, <a href="/A372033/b372033.txt">Table of n, a(n) for n = 0..1000</a>
%H A372033 Wikipedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">Motzkin number</a>
%F A372033 a(n) = A001006(n) + A333498(n).
%p A372033 b:= proc(x, y, h) option remember; `if`(x=0, h+1, add(
%p A372033       b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
%p A372033     end:
%p A372033 a:= n-> b(n, 0$2):
%p A372033 seq(a(n), n=0..35);
%Y A372033 Row sums of A372014.
%Y A372033 Cf. A001006, A261003, A333498.
%K A372033 nonn
%O A372033 0,3
%A A372033 _Alois P. Heinz_, Apr 16 2024