A136439 -id:A136439 - cp's OEIS frontend

A261003 a(n) = A136439(n) + Catalan(n).

1, 2, 5, 15, 48, 160, 549, 1924, 6851, 24700, 89945, 330239, 1220884, 4540128, 16968958, 63701573, 240059998, 907760348, 3443048256, 13094812968, 49925646786, 190772846082, 730451716847, 2802033270234, 10767028435468, 41438212118088, 159711845145544, 616393788920923, 2381898673172602

Offset: 0

N. J. A. Sloane, Aug 12 2015

nonn

This is the quantity S_{n-1} as given by a literal reading of Dershowitz and Rinderknecht (2015), Equations (1) and (2). If the lower limit in the right-hand sum in Eq. (1) is changed to "h >= 2", we obtain A136439.

a(n) is the total number of levels visited by all Dyck paths of semilength n. - Alois P. Heinz, Apr 14 2024

Alois P. Heinz, Table of n, a(n) for n = 0..650
N. Dershowitz and C. Rinderknecht, The Average Height of Catalan Trees by Counting Lattice Paths, Math. Mag., 88 (No. 3, 2015), 187-195.

Cf. A000108, A136439.

Row sums of A371928.

# Maple code for Equations (1) and (2) of Dershowitz and Rinderknecht (2015).
H:=proc(n,h) local b,k; b:=binomial; add(b(2*n,n+1-k*h)-2*b(2*n,n-k*h)+b(2*n,n-1-k*h),k=1..n+1); end;
S1:=n->add(H(n,h),h=1..n+1); [seq(S1(n),n=0..30)];

b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[If[x+j > y, b[x-1, y-j, Max[h, y-j]], 0], {j, Range[-1, Min[1, y]]~Complement~{0}}]];
a[n_] :=  b[2n, 0, 0] + CatalanNumber[n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 15 2023, after Alois P. Heinz in A136439 *)

a(n) = A000108(n) + A136439(n).

A333498 Sum of the heights of all Motzkin paths of length n.

Original entry on oeis.org

0, 0, 1, 3, 9, 25, 70, 196, 552, 1560, 4423, 12573, 35826, 102310, 292786, 839554, 2411945, 6941593, 20011328, 57779038, 167069317, 483739961, 1402413161, 4070537585, 11827842021, 34403798725, 100167396088, 291903951462, 851380987390, 2485175809878

Offset: 0

Alois P. Heinz, Mar 24 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Motzkin number

Cf. A001006, A057585, A097862, A136439, A333504, A333608.

b:= proc(x, y, h) option remember; `if`(x=0, h, add(
      b(x-1, y+j, max(h, y)), j=-min(1, y)..min(1, x-y-1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..35);

b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[b[x - 1, y + j, Max[h, y]], {j, -Min[1, y], Min[1, x - y - 1]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, May 10 2020, after Maple *)

a(n) = Sum_{k=1..floor(n/2)} k * A097862(n,k).

cp's OEIS Frontend

A261003 a(n) = A136439(n) + Catalan(n).

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