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