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A289054 Number of Dyck paths having exactly two peaks in each of the levels 1,...,n and no other peaks.

%I A289054 #10 Jul 06 2017 09:27:24
%S A289054 1,1,9,471,82899,36913581,34878248649,62045165964951,
%T A289054 190543753640526939,945931782247964900901,7209377339218632463758129,
%U A289054 80920117567254715984058542191,1292645840976784584918218615760819,28557854803885245556927129118200208781
%N A289054 Number of Dyck paths having exactly two peaks in each of the levels 1,...,n and no other peaks.
%C A289054 The semilengths of Dyck paths counted by a(n) are elements of the integer interval [3*n-1, n*(n+1)] for n>0.
%H A289054 Alois P. Heinz, <a href="/A289054/b289054.txt">Table of n, a(n) for n = 0..60</a>
%H A289054 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A289054 . a(2) = 9:           /\/\        /\/\        /\/\             /\  /\
%e A289054 .                /\/\/    \    /\/    \/\    /    \/\/\   /\/\/  \/  \
%e A289054 .
%e A289054 .    /\    /\      /\  /\      /\      /\    /\    /\      /\  /\
%e A289054 . /\/  \/\/  \  /\/  \/  \/\  /  \/\/\/  \  /  \/\/  \/\  /  \/  \/\/\ .
%p A289054 b:= proc(n, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
%p A289054       `if`(v<2, 0, add(b(n-j, i, v-1)*(binomial(i, 2)*
%p A289054        binomial(j-1, i-3)), i=1..min(j+2, n-j))))
%p A289054     end:
%p A289054 a:= n-> `if`(n=0, 1, add(b(w, 2, n), w=3*n-1..n*(n+1))):
%p A289054 seq(a(n), n=0..15);
%t A289054 b[n_, j_, v_]:=b[n, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, i, v - 1] Binomial[i, 2] Binomial[j - 1, i - 3], {i, Min[j + 2, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, 2, n], {w, 3*n - 1, n(n + 1)}]]; Table[a[n], {n, 0, 15}] (* _Indranil Ghosh_, Jul 06 2017, after Maple code *)
%Y A289054 Column k=2 of A288972.
%K A289054 nonn
%O A289054 0,3
%A A289054 _Alois P. Heinz_, Jun 23 2017