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A289030 Number of Dyck paths having exactly n peaks in each of the levels 1,2,3 and no other peaks.

%I A289030 #16 Jul 07 2017 06:03:33
%S A289030 1,10,471,27076,1713955,114751470,7969151855,567878871304,
%T A289030 41247976697019,3040572724077010,226777538499783271,
%U A289030 17076122335343354700,1296037531424347164115,99025149551454886937590,7609414766853344476768095,587623058661705739915402256
%N A289030 Number of Dyck paths having exactly n peaks in each of the levels 1,2,3 and no other peaks.
%C A289030 The semilengths of Dyck paths counted by a(n) are elements of the integer interval [3*n+2, 6*n] for n>0.
%H A289030 Alois P. Heinz, <a href="/A289030/b289030.txt">Table of n, a(n) for n = 0..523</a>
%H A289030 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A289030 . a(1) = 10:
%e A289030 .
%e A289030 .        /\        /\          /\        /\
%e A289030 .     /\/  \      /  \/\    /\/  \      /  \/\
%e A289030 .  /\/      \  /\/      \  /      \/\  /      \/\
%e A289030 .
%e A289030 .                /\        /\                /\
%e A289030 .           /\  /  \      /  \  /\    /\    /  \
%e A289030 .        /\/  \/    \  /\/    \/  \  /  \/\/    \
%e A289030 .
%e A289030 .              /\        /\            /\
%e A289030 .         /\  /  \      /  \    /\    /  \  /\
%e A289030 .        /  \/    \/\  /    \/\/  \  /    \/  \/\  .
%p A289030 b:= proc(n, k, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
%p A289030       `if`(v<2, 0, add(b(n-j, k, i, v-1)*(binomial(i, k)*
%p A289030        binomial(j-1, i-1-k)), i=1..min(j+k, n-j))))
%p A289030     end:
%p A289030 a:= n-> `if`(n=0, 1, add(b(w, n$2, 3), w=3*n+2..6*n)):
%p A289030 seq(a(n), n=0..15);
%t A289030 b[n_, k_, j_, v_]:=b[n, k, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, k, i, v - 1] Binomial[i, k] Binomial[j - 1, i - 1 - k], {i, Min[j + k, n - j]}]]]; a[n_]:=If[n==0, 1, Sum[b[w, n, n, 3], {w, 3n + 2, 6n}]]; Table[a[n], {n, 0, 15}] (* _Indranil Ghosh_, Jul 06 2017, after maple code *)
%Y A289030 Row n=3 of A288972.
%K A289030 nonn
%O A289030 0,2
%A A289030 _Alois P. Heinz_, Jun 22 2017