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%I #12 May 24 2020 16:20:40
%S 1,1,19,84,461,2222,10577,48943,222627,997735,4417674,19359659,
%T 84099436,362570722,1552681071,6609823112,27989970166,117967914457,
%U 495087382572,2069827499508,8623283249034,35811917284318,148289870077879,612382134256433,2522591250558641
%N Number of Dyck paths of semilength n such that the maximal number of peaks per level equals eight.
%H Alois P. Heinz, <a href="/A288749/b288749.txt">Table of n, a(n) for n = 8..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%p b:= proc(n, k, j) option remember; `if`(j=n, 1, add(
%p b(n-j, k, i)*add(binomial(i, m)*binomial(j-1, i-1-m),
%p m=max(0, i-j)..min(k, i-1)), i=1..min(j+k, n-j)))
%p end:
%p g:= proc(n, k) option remember; add(b(n, k, j), j=1..k) end:
%p a:= n-> g(n, 8)-g(n, 7):
%p seq(a(n), n=8..35);
%t b[n_, k_, j_]:=b[n, k, j]=If[j==n, 1, Sum[b[n - j,k, i] Sum[Binomial[i, m] Binomial[j - 1, i - 1 - m], {m, Max[0, i - j], Min[k, i - 1]}], {i, Min[j + k, n - j]}]]; g[n_, k_]:=Sum[b[n, k, j], {j, k}]; Table[g[n, 8] - g[n, 7], {n, 8, 35}] (* _Indranil Ghosh_, Aug 08 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(b(n - j, k, i)*sum(binomial(i, m)*binomial(j - 1, i - 1 - m) for m in range(max(0, i - j), min(k, i - 1) + 1)) for i in range(1, min(j + k, n - j) + 1))
%o def g(n, k): return sum(b(n, k, j) for j in range(1, k + 1))
%o def a(n): return g(n, 8) - g(n, 7)
%o print([a(n) for n in range(8, 36)]) # _Indranil Ghosh_, Aug 08 2017
%Y Column k=8 of A287822.
%Y Cf. A000108.
%K nonn
%O 8,3
%A _Alois P. Heinz_, Jun 14 2017
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