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A288683 Number of Dyck paths of semilength n such that no positive level has fewer than seven peaks.

%I A288683 #13 Apr 24 2021 08:44:55
%S A288683 1,0,0,0,0,0,0,1,1,1,1,1,1,1,1,9,234,2350,12567,44971,127475,320491,
%T A288683 756677,1720610,3821223,8436508,19793620,59810128,268048977,
%U A288683 1458971589,7720465569,36927931597,159094351283,626621217546,2296016964863,7949275945740
%N A288683 Number of Dyck paths of semilength n such that no positive level has fewer than seven peaks.
%H A288683 Alois P. Heinz, <a href="/A288683/b288683.txt">Table of n, a(n) for n = 0..300</a>
%H A288683 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%t A288683 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, 7, j], {j, 7, n}]]; Table[a[n], {n, 0, 35}] (* _Indranil Ghosh_, Aug 10 2017 *)
%o A288683 (Python)
%o A288683 from sympy.core.cache import cacheit
%o A288683 from sympy import binomial
%o A288683 @cacheit
%o A288683 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 A288683 def a(n): return 1 if n==0 else sum([b(n, 7, j) for j in range(7, n + 1)])
%o A288683 print([a(n) for n in range(36)]) # _Indranil Ghosh_, Aug 10 2017
%Y A288683 Column k=7 of A288386.
%Y A288683 Cf. A000108.
%K A288683 nonn
%O A288683 0,16
%A A288683 _Alois P. Heinz_, Jun 13 2017