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%I #13 Apr 24 2021 08:45:03
%S 1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,11,462,7217,57783,289400,
%T 1043781,3042593,7833174,18821247,43417043,97550980,215243289,
%U 469069428,1020806036,2342090587,6886047798,32238887181,199504672863,1232775909721,6881782444707
%N Number of Dyck paths of semilength n such that no positive level has fewer than nine peaks.
%H Alois P. Heinz, <a href="/A288685/b288685.txt">Table of n, a(n) for n = 0..300</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%t 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, 9, j], {j, 9, n}]]; Table[a[n], {n, 0, 40}] (* _Indranil Ghosh_, Aug 10 2017 *)
%o (Python)
%o from sympy.core.cache import cacheit
%o from sympy import binomial
%o @cacheit
%o 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 def a(n): return 1 if n==0 else sum([b(n, 9, j) for j in range(9, n + 1)])
%o print([a(n) for n in range(41)]) # _Indranil Ghosh_, Aug 10 2017
%Y Column k=9 of A288386.
%Y Cf. A000108.
%K nonn
%O 0,20
%A _Alois P. Heinz_, Jun 13 2017
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