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%I #16 May 29 2018 03:49:30
%S 1,1,1,1,3,6,12,31,68,186,506,1299,3481,9712,27692,79587,232743,
%T 694896,2086245,6248158,18771510,57007483,175149700,542313513,
%U 1688360997,5288335561,16679137617,52933231538,168768966207,539981776609,1733555552587,5587076558809
%N Number of Dyck paths of semilength n such that the number of peaks is strongly increasing from lower to higher levels and no positive level is peakless.
%H Alois P. Heinz, <a href="/A288147/b288147.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) = 6:
%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(1, i-j)..min(k-1, 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..34);
%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[1, i - j], Min[k - 1, 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, 34}] (* _Jean-François Alcover_, May 29 2018, from Maple *)
%Y Cf. A000108, A008930, A048285, A288140, A288141, A288146.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Jun 05 2017
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