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A316728 Number T(n,k) of permutations of {0,1,...,2n} with first element k whose sequence of ascents and descents forms a Dyck path; triangle T(n,k), n>=0, 0<=k<=2n, read by rows.

%I A316728 #27 Mar 27 2021 08:07:52
%S A316728 1,1,1,0,8,7,5,2,0,172,150,121,87,52,22,0,7296,6440,5464,4411,3337,
%T A316728 2306,1380,604,0,518324,463578,405024,344260,283073,223333,166856,
%U A316728 115250,69772,31238,0,55717312,50416894,44928220,39348036,33777456,28318137,23068057,18117190,13543456,9409366,5759740,2620708,0
%N A316728 Number T(n,k) of permutations of {0,1,...,2n} with first element k whose sequence of ascents and descents forms a Dyck path; triangle T(n,k), n>=0, 0<=k<=2n, read by rows.
%H A316728 Alois P. Heinz, <a href="/A316728/b316728.txt">Rows n = 0..100, flattened</a>
%H A316728 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A316728 Sum_{k=0..2n} T(n,k) = T(n+1,2n+1) = A177042(n).
%F A316728 Sum_{k=0..2n} (k+1) * T(n,k) = A079484(n).
%e A316728 T(2,0) = 8: 01432, 02143, 02431, 03142, 03241, 03421, 04132, 04231.
%e A316728 T(2,1) = 7: 12043, 12430, 13042, 13240, 13420, 14032, 14230.
%e A316728 T(2,2) = 5: 23041, 23140, 23410, 24031, 24130.
%e A316728 T(2,3) = 2: 34021, 34120.
%e A316728 T(2,4) = 0.
%e A316728 Triangle T(n,k) begins:
%e A316728        1;
%e A316728        1,      1,      0;
%e A316728        8,      7,      5,      2,      0;
%e A316728      172,    150,    121,     87,     52,     22,      0;
%e A316728     7296,   6440,   5464,   4411,   3337,   2306,   1380,    604,     0;
%e A316728   518324, 463578, 405024, 344260, 283073, 223333, 166856, 115250, 69772, 31238, 0;
%p A316728 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A316728       `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
%p A316728       `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
%p A316728     end:
%p A316728 T:= (n, k)-> b(k, 2*n-k, 0):
%p A316728 seq(seq(T(n, k), k=0..2*n), n=0..8);
%t A316728 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
%t A316728      If[t > 0,     Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +
%t A316728      If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
%t A316728 T[n_, k_] := b[k, 2n - k, 0];
%t A316728 Table[Table[T[n, k], {k, 0, 2n}], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Mar 27 2021, after _Alois P. Heinz_ *)
%Y A316728 Column k=0 gives A303285.
%Y A316728 Row sums and T(n+1,2n+1) give A177042.
%Y A316728 T(n,n) gives A316727.
%Y A316728 T(n+1,n) gives A316730.
%Y A316728 T(n,2n) gives A000007.
%Y A316728 Cf. A079484.
%K A316728 nonn,tabf
%O A316728 0,5
%A A316728 _Alois P. Heinz_, Jul 11 2018