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A288686 Number of Dyck paths of semilength n such that no positive level has fewer than ten peaks.

%I A288686 #13 May 12 2020 04:44:45
%S A288686 1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,12,618,11695,112434,665219,
%T A288686 2756389,8890492,24410518,60972735,144203914,329766287,737405644,
%U A288686 1623087349,3531560786,7633789153,16745585892,41482511559,152244106469,886899776271
%N A288686 Number of Dyck paths of semilength n such that no positive level has fewer than ten peaks.
%H A288686 Alois P. Heinz, <a href="/A288686/b288686.txt">Table of n, a(n) for n = 0..300</a>
%H A288686 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%t A288686 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, 10, j], {j, 10, n}]]; Table[a[n], {n, 0, 40}] (* _Indranil Ghosh_, Aug 10 2017 *)
%o A288686 (Python)
%o A288686 from sympy.core.cache import cacheit
%o A288686 from sympy import binomial
%o A288686 @cacheit
%o A288686 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 A288686 def a(n): return 1 if n==0 else sum(b(n, 10, j) for j in range(10, n + 1))
%o A288686 print([a(n) for n in range(41)]) # _Indranil Ghosh_, Aug 10 2017
%Y A288686 Column k=10 of A288386.
%Y A288686 Cf. A000108.
%K A288686 nonn
%O A288686 0,22
%A A288686 _Alois P. Heinz_, Jun 13 2017