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A288678 Number of Dyck paths of semilength n such that no positive level has fewer than two peaks.

%I A288678 #14 Jun 04 2020 02:54:50
%S A288678 1,0,1,1,1,4,14,35,91,268,864,2833,9279,30670,102975,351148,1212886,
%T A288678 4232714,14900843,52865511,188871400,679029570,2455099043,8922220725,
%U A288678 32576194260,119447959183,439700905503,1624436294053,6021371511844,22388679839583,83484414608203
%N A288678 Number of Dyck paths of semilength n such that no positive level has fewer than two peaks.
%H A288678 Alois P. Heinz, <a href="/A288678/b288678.txt">Table of n, a(n) for n = 0..300</a>
%H A288678 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%t A288678 b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[k, i - j], i - 1}] b[n - j, k, i], {i, n - j}]];  a[n_]:=If[n==0, 1, Sum[b[n, 2, j], {j, 2, n}]];Table[a[n], {n, 0, 30}] (* _Indranil Ghosh_, Aug 09 2017 *)
%o A288678 (Python)
%o A288678 from sympy.core.cache import cacheit
%o A288678 from sympy import binomial
%o A288678 @cacheit
%o A288678 def b(n, k, j): return 1 if j==n else sum(sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(k, i - j), i))*b(n - j, k, i) for i in range(1, n - j + 1))
%o A288678 def a(n): return 1 if n==0 else sum(b(n, 2, j) for j in range(2, n + 1))
%o A288678 print([a(n) for n in range(31)]) # _Indranil Ghosh_, Aug 09 2017
%Y A288678 Column k=2 of A288386.
%Y A288678 Cf. A000108.
%K A288678 nonn
%O A288678 0,6
%A A288678 _Alois P. Heinz_, Jun 13 2017