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A289020 Number of Dyck paths having exactly one peak in each of the levels 1,...,n and no other peaks.

%I A289020 #28 Jul 07 2017 06:03:13
%S A289020 1,1,2,10,92,1348,28808,845800,32664944,1605553552,97868465696,
%T A289020 7245440815264,640359291096512,66598657958731840,8051483595083729024,
%U A289020 1119653568781387712128,177465810459239319017216,31804047327185301634148608,6398867435594240638421950976
%N A289020 Number of Dyck paths having exactly one peak in each of the levels 1,...,n and no other peaks.
%C A289020 The semilengths of Dyck paths counted by a(n) are elements of the integer interval [2*n-1, n*(n+1)/2] = [A060747(n), A000217(n)] for n>0.
%H A289020 Alois P. Heinz, <a href="/A289020/b289020.txt">Table of n, a(n) for n = 0..100</a>
%H A289020 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A289020 . a(2) = 2:      /\    /\
%e A289020 .             /\/  \  /  \/\  .
%p A289020 b:= proc(n, j, v) option remember; `if`(n=j,
%p A289020       `if`(v=1, 1, 0), `if`(v<2, 0, add(b(n-j, i, v-1)*
%p A289020        i*binomial(j-1, i-2), i=1..min(j+1, n-j))))
%p A289020     end:
%p A289020 a:= n-> `if`(n=0, 1, add(b(w, 1, n), w=2*n-1..n*(n+1)/2)):
%p A289020 seq(a(n), n=0..18);
%t A289020 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]*i*Binomial[j - 1, i - 2], {i, Min[j + 1, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, 1, n], {w, 2*n - 1, n*(n + 1)/2}]]; Table[a[n], {n, 0, 18}] (* _Indranil Ghosh_, Jul 06 2017, after Maple code *)
%Y A289020 Column k=1 of A288972.
%Y A289020 Cf. A000217, A060747, A281874, A287846.
%K A289020 nonn
%O A289020 0,3
%A A289020 _Alois P. Heinz_, Jun 22 2017