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%I #11 Jun 02 2018 10:35:53
%S 1,1,1,1,6,10,27,84,226,770,2390,7579,25222,84299,285284,976105,
%T 3386494,11858759,41782516,148205047,529101609,1899680494,6854597493,
%U 24847293152,90460431604,330654288724,1213033321450,4465027739962,16486012746085,61044028354833
%N Number of Dyck paths of semilength n such that all positive levels up to the highest level have a positive number of peaks and the number of peaks of adjacent levels is different.
%H Alois P. Heinz, <a href="/A287989/b287989.txt">Table of n, a(n) for n = 0..150</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e . (4) = 6:
%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(n-j, i-1)} minus {k}), 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..30);
%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, Range[Max[1, i - j], Min[n - j, i - 1]] ~Complement~ {k}}], {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, 30}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y Cf. A000108, A287993.
%K nonn
%O 0,5
%A _Alois P. Heinz_, Jun 04 2017
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