cp's OEIS Frontend

A288940 Number of Dyck paths having n (positive) levels and exactly n peaks per level.

%I A288940 #19 Jul 07 2017 06:03:48
%S A288940 1,1,9,27076,147556480375,4711342006036190504484,
%T A288940 2162932174406679548553402518043252929,
%U A288940 29605698225102450501737027784037791564430800582087459328,22346336234943531646124131709622442581521043809236751640919325993842966011809319
%N A288940 Number of Dyck paths having n (positive) levels and exactly n peaks per level.
%C A288940 The semilengths of Dyck paths counted by a(n) are elements of the integer interval [n^2+n-1, n^2*(n+1)/2] for n>0.
%H A288940 Alois P. Heinz, <a href="/A288940/b288940.txt">Table of n, a(n) for n = 0..15</a>
%H A288940 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A288940 . a(1) = 1:       /\  .
%e A288940 .
%e A288940 . a(2) = 9:           /\/\        /\/\        /\/\             /\  /\
%e A288940 .                /\/\/    \    /\/    \/\    /    \/\/\   /\/\/  \/  \
%e A288940 .
%e A288940 .    /\    /\      /\  /\      /\      /\    /\    /\      /\  /\
%e A288940 . /\/  \/\/  \  /\/  \/  \/\  /  \/\/\/  \  /  \/\/  \/\  /  \/  \/\/\ .
%p A288940 b:= proc(n, k, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
%p A288940       `if`(v<2, 0, add(b(n-j, k, i, v-1)*(binomial(i, k)
%p A288940        *binomial(j-1, i-1-k)), i=1..min(j+k, n-j))))
%p A288940     end:
%p A288940 a:= n-> `if`(n=0, 1, add(b(k, n$3), k=n^2+n-1..n^2*(n+1)/2)):
%p A288940 seq(a(n), n=0..7);
%t A288940 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[k, n, n, n], {k, n^2 + n - 1, n^2*(n + 1)/2}]]; Table[a[n], {n, 0, 8}] (* _Indranil Ghosh_, Jul 06 2017, after Maple code *)
%Y A288940 Main diagonal of A288972.
%Y A288940 Cf. A288318.
%K A288940 nonn
%O A288940 0,3
%A A288940 _Alois P. Heinz_, Jun 19 2017