A288748 - OEIS

%I #13 Apr 24 2021 08:45:18

%S 1,1,17,71,368,1697,7769,34751,153313,668088,2882104,12329145,

%T 52358300,220901081,926638057,3867432363,16068748557,66495876593,

%U 274178902925,1126793986670,4616878543095,18864740697016,76885237242318,312611605360287,1268261191750753

%N Number of Dyck paths of semilength n such that the maximal number of peaks per level equals seven.

%H Alois P. Heinz, <a href="/A288748/b288748.txt">Table of n, a(n) for n = 7..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, 7)-g(n, 6):

%p seq(a(n), n=7..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, 7] - g[n, 6], {n, 7, 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, 7) - g(n, 6)

%o print([a(n) for n in range(7, 36)]) # _Indranil Ghosh_, Aug 08 2017

%Y Column k=7 of A287822.

%Y Cf. A000108.

%K nonn

%O 7,3

%A _Alois P. Heinz_, Jun 14 2017