A288684 - OEIS

%I #13 Apr 24 2021 08:44:59

%S 1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,10,335,4241,27915,117971,373845,

%T 1002089,2456082,5725439,12935530,28622833,62588817,139046970,

%U 353173119,1305216091,7035422989,41539474198,227550374938,1115122502718,4917988882292

%N Number of Dyck paths of semilength n such that no positive level has fewer than eight peaks.

%H Alois P. Heinz, <a href="/A288684/b288684.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, 8, j], {j, 8, 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, 8, j) for j in range(8, n + 1)])

%o print([a(n) for n in range(41)]) # _Indranil Ghosh_, Aug 10 2017

%Y Column k=8 of A288386.

%Y Cf. A000108.

%K nonn

%O 0,18

%A _Alois P. Heinz_, Jun 13 2017