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A300954 Number of Dyck paths whose sequence of ascent lengths is exactly n+1, n+2, ..., 2n.

%I A300954 #19 Jun 02 2018 10:38:19
%S A300954 1,1,3,26,425,10647,365512,16067454,864721566,55202528425,
%T A300954 4083666929771,343854336973368,32493430569907125,3406873823160467912,
%U A300954 392619681705581846700,49342834390595374213214,6717520607597479710109299,984991858956314599670220717,154785386247352261724279606367
%N A300954 Number of Dyck paths whose sequence of ascent lengths is exactly n+1, n+2, ..., 2n.
%C A300954 Dyck paths counted by a(n) have semilength (3*n^2 + n)/2 = A005449(n) and length A049451(n).
%H A300954 Alois P. Heinz, <a href="/A300954/b300954.txt">Table of n, a(n) for n = 0..150</a>
%H A300954 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A300954 a(n) = A107876(2n,n).
%e A300954 a(0) = 1: the empty path.
%e A300954 a(1) = 1: uudd.
%e A300954 a(2) = 3: uuuduuuudddddd, uuudduuuuddddd, uuuddduuuudddd.
%p A300954 a:= proc(m) option remember; local b; b:=
%p A300954       proc(n, i) option remember; `if`(i>=2*m, 1,
%p A300954         add(b(n+i-j, i+1), j=1..n+i))
%p A300954       end; b(0, m+1)
%p A300954     end:
%p A300954 seq(a(n), n=0..20);
%t A300954 a[m_] := a[m] = Module[{b}, b[n_, i_] := b[n, i] = If[i >= 2m, 1, Sum[b[n + i - j, i + 1], {j, 1, n + i}]]; b[0, m + 1]];
%t A300954 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 02 2018, from Maple *)
%Y A300954 Main diagonal of A107876.
%Y A300954 Cf. A005449, A049451.
%K A300954 nonn
%O A300954 0,3
%A A300954 _Alois P. Heinz_, Mar 16 2018