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A287987 Number of Dyck paths of semilength n such that all positive levels have the same number of peaks.

%I A287987 #17 May 24 2018 06:18:12
%S A287987 1,1,1,3,1,8,13,13,54,132,280,547,1219,3904,11107,25082,53777,137751,
%T A287987 419831,1257599,3453557,8911341,22636845,59890162,172264224,529706648,
%U A287987 1630328686,4765347773,13125989799,35253234315,97531470556,287880507391,894915519516
%N A287987 Number of Dyck paths of semilength n such that all positive levels have the same number of peaks.
%H A287987 Alois P. Heinz, <a href="/A287987/b287987.txt">Table of n, a(n) for n = 0..500</a>
%H A287987 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A287987 . a(3) = 3:                         /\        /\
%e A287987 .                    /\/\/\      /\/  \      /  \/\  .
%e A287987 .
%e A287987 . a(5) = 8:
%e A287987 .                       /\/\      /\/\      /\/\
%e A287987 .      /\/\/\/\/\  /\/\/    \  /\/    \/\  /    \/\/\
%e A287987 .
%e A287987 .            /\        /\          /\        /\
%e A287987 .         /\/  \      /  \/\    /\/  \      /  \/\
%e A287987 .      /\/      \  /\/      \  /      \/\  /      \/\  .
%p A287987 b:= proc(n, k, j) option remember; `if`(n=j, 1,
%p A287987        add(binomial(i, k)*binomial(j-1, i-1-k)
%p A287987          *b(n-j, k, i), i=1+k..min(j+k, n-j)))
%p A287987     end:
%p A287987 a:= n-> 1+add(b(n, j$2), j=1..n/2):
%p A287987 seq(a(n), n=0..33);
%t A287987 b[n_, k_, j_] := b[n, k, j] = If[n == j, 1, Sum[Binomial[i, k]*Binomial[j - 1, i - 1 - k]*b[n - j, k, i], {i, 1 + k, Min[j + k, n - j]}]];
%t A287987 a[n_] := 1 + Sum[b[n, j, j], {j, 1, n/2}];
%t A287987 Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, May 24 2018, translated from Maple *)
%Y A287987 Row sums of A288318.
%Y A287987 Cf. A000108, A287845, A287846, A287993, A288109.
%K A287987 nonn
%O A287987 0,4
%A A287987 _Alois P. Heinz_, Jun 03 2017