cp's OEIS Frontend

A287845 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly two peaks.

%I A287845 #28 Mar 04 2024 15:01:47
%S A287845 1,0,1,0,0,3,6,0,9,54,138,207,360,1368,4545,11304,25182,61605,173916,
%T A287845 498798,1347417,3497328,9147060,24630669,67414590,184065966,498495303,
%U A287845 1345622436,3642036804,9900361107,26982011250,73570082760,200540053395,546660151722
%N A287845 Number of Dyck paths of semilength n such that each positive level up to the highest nonempty level has exactly two peaks.
%H A287845 Alois P. Heinz, <a href="/A287845/b287845.txt">Table of n, a(n) for n = 0..1000</a>
%H A287845 Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 15.
%H A287845 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A287845 . a(2) = 1:   /\/\  .
%e A287845 .
%e A287845 . a(5) = 3:
%e A287845 .
%e A287845 .               /\/\     /\/\     /\/\
%e A287845 .          /\/\/    \ /\/    \/\ /    \/\/\ .
%p A287845 b:= proc(n, j) option remember;
%p A287845       `if`(n=j or n=0, 1, add(b(n-j, i)*i*(i-1)/2
%p A287845        *binomial(j-1, i-3), i=3..min(j+2, n-j)))
%p A287845     end:
%p A287845 a:= n-> b(n, 2):
%p A287845 seq(a(n), n=0..35);
%t A287845 b[n_, j_] := b[n, j] = If[n == j || n == 0, 1, Sum[b[n - j, i]*i*(i - 1)/2* Binomial[j - 1, i - 3], {i, 3, Min[j + 2, n - j]}]];
%t A287845 a[n_] := b[n, 2];
%t A287845 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 25 2018, translated from Maple *)
%Y A287845 Column k=2 of A288318.
%Y A287845 Cf. A000108, A281874, A287843, A287846, A287963, A287987.
%K A287845 nonn
%O A287845 0,6
%A A287845 _Alois P. Heinz_, Jun 01 2017