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A287843 Number of Dyck paths of semilength n such that each level with peaks has exactly two peaks.

%I A287843 #22 May 25 2018 08:28:43
%S A287843 1,0,1,1,2,5,15,27,76,196,548,1388,3621,9894,27553,75346,205634,
%T A287843 563729,1565409,4370226,12191929,33980329,94874987,265668404,
%U A287843 745652478,2095025688,5889310438,16565399257,46633521554,131388795335,370434641340,1044917168292
%N A287843 Number of Dyck paths of semilength n such that each level with peaks has exactly two peaks.
%H A287843 Alois P. Heinz, <a href="/A287843/b287843.txt">Table of n, a(n) for n = 0..1000</a>
%H A287843 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A287843 . a(2) = 1:   /\/\ .
%e A287843 .
%e A287843 . a(3) = 1:   /\/\
%e A287843 .            /    \ .
%e A287843 .
%e A287843 . a(4) = 2:              /\/\
%e A287843 .            /\  /\     /    \
%e A287843 .           /  \/  \   /      \ .
%e A287843 .
%e A287843 . a(5) = 5:                                               /\/\
%e A287843 .                                             /\  /\     /    \
%e A287843 .               /\/\     /\/\     /\/\       /  \/  \   /      \
%e A287843 .          /\/\/    \ /\/    \/\ /    \/\/\ /        \ /        \ .
%p A287843 b:= proc(n, j) option remember; `if`(n=j or n=0, 1,
%p A287843       add(b(n-j, i)*(binomial(j-1, i-1) +i*(i-1)/2*
%p A287843       binomial(j-1, i-3)), i=1..min(j+3, n-j)))
%p A287843     end:
%p A287843 a:= n-> b(n, 2):
%p A287843 seq(a(n), n=0..35);
%t A287843 b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*(Binomial[j - 1, i-1] + i*(i-1)/2*Binomial[j-1, i-3]), {i, 1, Min[j + 3, n - j]}]];
%t A287843 a[n_] := b[n, 2];
%t A287843 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 25 2018, translated from Maple *)
%Y A287843 Cf. A000108, A281874, A287845, A287846, A287963.
%Y A287843 Column k=2 of A288108.
%K A287843 nonn
%O A287843 0,5
%A A287843 _Alois P. Heinz_, Jun 01 2017