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%I #17 May 29 2018 03:04:32
%S 1,1,1,3,4,12,28,63,177,455,1233,3383,9359,26809,77078,223201,653982,
%T 1934508,5783712,17431660,52879184,161386859,495432345,1530191918,
%U 4754079840,14849407892,46604383972,146897291083,464892421363,1477052536749,4711124635655
%N Number of Dyck paths of semilength n such that the number of peaks is weakly decreasing from lower to higher levels.
%H Alois P. Heinz, <a href="/A288140/b288140.txt">Table of n, a(n) for n = 0..200</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e . a(5) = 12:
%e . /\ /\ /\
%e . /\/\/\/\/\ /\/\/\/ \ /\/\/ \/\ /\/ \/\/\
%e .
%e . /\ /\/\ /\/\ /\/\
%e . / \/\/\/\ /\/\/ \ /\/ \/\ / \/\/\
%e .
%e . /\ /\ /\ /\
%e . /\/ \ / \/\ /\/ \ / \/\
%e . /\/ \ /\/ \ / \/\ / \/\ .
%p b:= proc(n, k, j) option remember; `if`(n=j, 1, add(add(
%p b(n-j, t, i)*binomial(i, t)*binomial(j-1, i-1-t),
%p t=max(k, i-j)..min(n-j, i-1)), i=1..n-j))
%p end:
%p a:= n-> `if`(n=0, 1, add(b(n, k$2), k=1..n)):
%p seq(a(n), n=0..31);
%t b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Sum[b[n - j, t, i]* Binomial[i, t]*Binomial[j - 1, i - 1 - t], {t, Max[k, i - j], Min[n - j, i - 1]}], {i, 1, n - j}]];
%t a[n_] := If[n == 0, 1, Sum[b[n, k, k], {k, 1, n}]];
%t Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, May 29 2018, from Maple *)
%Y Cf. A000108, A008930, A048285, A288141, A288146, A288147.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Jun 05 2017
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