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A288679 Number of Dyck paths of semilength n such that no positive level has fewer than three peaks.

%I A288679 #14 Apr 24 2021 08:44:40
%S A288679 1,0,0,1,1,1,1,5,30,96,245,592,1543,4884,17660,64495,226442,766937,
%T A288679 2558655,8590293,29408344,102893203,366035420,1314955687,4747101946,
%U A288679 17184305311,62359953380,226978626707,829122987011,3040369502702,11191473790567,41342469523031
%N A288679 Number of Dyck paths of semilength n such that no positive level has fewer than three peaks.
%H A288679 Alois P. Heinz, <a href="/A288679/b288679.txt">Table of n, a(n) for n = 0..300</a>
%H A288679 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%t A288679 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, 3, j], {j, 3, n}]];Table[a[n], {n, 0, 35}] (* _Indranil Ghosh_, Aug 09 2017 *)
%o A288679 (Python)
%o A288679 from sympy.core.cache import cacheit
%o A288679 from sympy import binomial
%o A288679 @cacheit
%o A288679 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 A288679 def a(n): return 1 if n==0 else sum([b(n, 3, j) for j in range(3, n + 1)])
%o A288679 print([a(n) for n in range(36)]) # _Indranil Ghosh_, Aug 09 2017
%Y A288679 Column k=3 of A288386.
%Y A288679 Cf. A000108.
%K A288679 nonn
%O A288679 0,8
%A A288679 _Alois P. Heinz_, Jun 13 2017