A005700 -id:A005700 - cp's OEIS frontend

A379034 Number of intervals in the lattice of Motzkin paths of length n.

1, 1, 3, 9, 36, 156, 754, 3886, 21239, 121411, 721189, 4423167, 27884979, 180023715, 1186646085, 7966608939, 54362492505, 376401432105, 2640553802523, 18745081044417, 134511480800046, 974773373273658, 7127937477283896, 52556513927004360, 390494071802647015

Offset: 0

Ludovic Schwob, Dec 14 2024

nonn

a(n) is the number of nested pairs of Motzkin paths of length n.

			The a(3) = 9 pairs of Motzkin paths are:
                     _
  ___   _/\   /\_   / \   _/\
  ___   ___   ___   ___   _/\
.
   _           _     _
  / \   /\_   / \   /_\
  _/\   /\_   /\_   / \

Alois P. Heinz, Table of n, a(n) for n = 0..1061

Cf. A001006 (number of Motzkin paths), A005700 (number of intervals in the lattice of Dyck paths), A379035 (number of intervals in the lattice of Schroeder paths).

a:= proc(n) option remember; `if`(n<3, [1$2, 3][n+1],
      (n*(7*n+13)*(n+3)*a(n-1)+3*(n+1)*(n-1)*(7*n+6)*a(n-2)
        -27*(n+1)*(n-1)*(n-2)*a(n-3))/((n+4)*(n+3)^2))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Dec 14 2024

from collections import defaultdict
def a_gen(n): #generator of terms a(0) to a(n)
    D = {(0,0):1}
    yield 1
    for k in range(n):
        D2 = defaultdict(int)
        for i,j in D:
            d = D[(i,j)]
            for a in range(-1,2):
                for b in range(-1,2):
                    if 0 <= i+a <= j+b <= n-k-1:
                        D2[(i+a,j+b)] += d
        D = D2
        yield D[(0,0)]

A379035 Number of intervals in the lattice of Schroeder paths of length 2*n.

Original entry on oeis.org

1, 3, 20, 201, 2574, 38740, 654334, 12050561, 237383562, 4935186778, 107233505624, 2417430607946, 56224895607144, 1343171657532430, 32841495349026802, 819503313717327041, 20820095694805722362, 537474654188020125882, 14075078600122360679520, 373373810133893669112710

Offset: 0

Ludovic Schwob, Dec 14 2024

nonn

a(n) is the number of nested pairs of Schroeder paths of length 2*n.

			The a(2) = 20 pairs of Schroeder paths are:
.                              __
  ____   __/\   /\__   /\/\   /  \
  ____   ____   ____   ____   ____
.
   /\                   __     /\
  /  \   __/\   /\/\   /  \   /  \
  ____   __/\   __/\   __/\   __/\
.
                 __     /\
  /\__   /\/\   /  \   /  \   /\/\
  /\__   /\__   /\__   /\__   /\/\
.
   __     /\     __     /\     /\
  /  \   /  \   /__\   /__\   //\\
  /\/\   /\/\   /  \   /  \   /  \

Alois P. Heinz, Table of n, a(n) for n = 0..661

Cf. A006318 (number of Schroeder paths), A005700 (number of intervals in the lattice of Dyck paths), A379034 (number of intervals in the lattice of Motzkin paths).

from collections import defaultdict
def a_gen(n): #generator of terms a(0) to a(n)
    D = {(0,0):1}
    for k in range(2*n+2):
        D2 = defaultdict(int)
        for i,j in D:
            d = D[(i,j)]
            mi,mj = (i-k)%2,(j-k)%2
            for a in range(-mi,mi+1):
                for b in range(-mj,mj+1):
                    if 0 <= i+a <= j+b <= 2*n-k+1:
                        D2[(i+a,j+b)] += d
        D = D2
        if k%2==1:
            yield D[(0,0)]

A181571 Third column of triangle in A179898.

Original entry on oeis.org

1, 6, 40, 300, 2475, 22022, 208208, 2068560, 21414900, 229523800, 2533942752, 28698821320, 332357673375, 3925129083750, 47167131780000, 575637606165600, 7123515376299300, 89266155250239000, 1131410294476020000, 14489559984061890000, 187330691180608155180, 2443121585638964379864

Offset: 1

N. J. A. Sloane, Jan 30 2011

nonn

Cf. A179898, A005700, A005701.

A136440 Sum of heights of all 2-watermelons with wall of length 2*n.

Original entry on oeis.org

3, 11, 60, 406, 3171, 27411, 255617, 2528613, 26224097, 282706396, 3147801820, 36022733951, 422047425238, 5046771514478, 61438059222438, 759851375725606, 9530872096367508, 121063493728881999, 1555352365759798758

Offset: 1

Steven Finch, Apr 02 2008

nonn

Consider the set of all pairs of nonintersecting Dyck excursions of length 2*n (nonnegative walks with jumps -1,+1). The lower path begins and ends at 0; the upper path begins and ends at 2. a(n) is the sum of heights of all such upper-Dyck excursions.

M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall, Elec. J. Combin. 14 (2007) R64.

Cf. A005700, A078920.

c[n_] := 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!)
s[n_,a_] := Sum[If[k < 1, 0, DivisorSigma[0,k]*Binomial[2*n,n+a-k]/Binomial[2*n,n]], {k,a-n,a+n}]
t[n_,a_,b_] := Sum[If[(j < 1) || (k < 1), 0, DivisorSigma[0,GCD[j,k]]*Binomial[2*n,n+a-j]*Binomial[2*n,n+b-k]/Binomial[2*n,n]^2], {j,a-n,a+n}, {k,b-n,b+n}]
f[n_] := (n^2+5*n+6)*(s[n,-3]+s[n,3])-(6*n^2+18*n)*(s[n,-2]+s[n,2])+(15*n^2+27*n+6)*(s[n,-1]+s[n,1])-(20*n^2+28*n+24)*s[n,0]
g[n_] := t[n,-2,-2]-t[n,-1,-3]-2*t[n,-1,-2]+t[n,-1,-1]+2*t[n,-1,0]-t[n,-1,3]+2*t[n,0,-3]-4*t[n,0,0]+2*t[n,0,3]-t[n,1,-3]-2*t[n,1,-2]+2*t[n,1,-1]+2*t[n,1,0]+t[n,1,1]-t[n,1,3]+2*t[n,2,-2]-2*t[n,2,-1]-2*t[n,2,1]+t[n,2,2]
h[n_] := ((n+1)*(n+2)/(12*(2*n+1)))*( (n+1)*(n+2)*(n+3)*g[n]+f[n] ) - 1
a[n_] := h[n]*c[n]

A336674 Number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1).

Original entry on oeis.org

1, 1, 5, 65, 1757, 87129, 7286709, 965911665, 193387756045, 56251615627273, 23021497112124901, 12903943243053179681, 9680994096074346690365, 9530338509606467082850745, 12099590059386455266220499477

Offset: 0

Alejandro H. Morales, Jul 29 2020

nonn

a(n) is also the number of semistandard Young tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1) such that the entries in row i are at most i for i=1,...,n+3.

a(n) is also the number of semistandard Young tableaux T of shape (n-1,n-2,...,1) such that j-i < T(i,j) <= n+3 for all cells (i,j).

			For n=2 the a(2)=5 semistandard Young tableaux of skew shape (5,4,3,2,1)/(1) are determined by their first column which are [1,2,3,4], [1,2,3,5], [1,2,4,5], [1,3,4,5], and [2,3,4,5]. Also, the a(2)=5 semistandard Young tableaux of shape (1) with entries between 0 and 5 are [1], [2], [3], [4], and [5]. Also, the a(3)=70-5=65 are the semistandard Young tableaux of shape (2,1) with entries at most 6 excluding the five tableaux whose entry in the first row and first column is 1: [[1,1],[2]], [[1,1],[3]], [[1,1],[4]], and [[1,1],[5]].

A. H. Morales and D. G. Zhu, On the Okounkov-Olshanski formula for standard tableaux of skew shape, arXiv:2007.05006 [math.CO], 2020.

A110501, A005700 gives the number of terms of the Naruse hook length formula for the same skew shape.

b := proc(n)
    return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n);
end proc:
a := proc(n)
    return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;
end proc:
seq(a(n),n=0..10);

def b(n):
    return 2*(-1)^n*(1-4^n)*bernoulli(2*n)/factorial(2*n) ;
def a(n):
    return factorial(2*n+4)*factorial(2*n+6)*(b(n+1)*b(n+3)-b(n+2)^2)/6;
[a(i) for i in range(10)]

a(n) = ((2*n+4)!*(2*n+6)!/3!)*(b(n+1)*b(n+3)-b(n+2)^2) where b(n)=A110501(n)/(2*n)!.

A338368 Triangle of number of 312-avoiding reduced valid hook configurations with n hooks that are on permutations with 2n + k points, for n=1 and 1<=k<=n.

Original entry on oeis.org

1, 3, 5, 14, 51, 42, 84, 485, 849, 462, 594, 4743, 13004, 14819, 6006, 4719, 48309, 182311, 322789, 271452, 87516, 40898, 511607, 2472322, 5999489, 7794646, 5182011, 13856

Offset: 1

Michel Marcus, Oct 23 2020

			Triangle begins
  1
  3 5
  14 51 42
  84 485 849 462
  594 4743 13004 14819 6006
  4719 48309 182311 322789 271452 87516
  40898 511607 2472322 5999489 7794646 5182011 13856

Ilani Axelrod-Freed, 312-Avoiding Reduced Valid Hook Configurations and Duck Words, arXiv:2010.11834 [math.CO], 2020.
Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019.

Cf. A005700 (1st column), A005789 (right diagonal).

A338403 Regular triangle read by rows: T(n,k) is the number of (n,k)-Duck words, for n>=1 and 0<=k<=n-1.

Original entry on oeis.org

1, 2, 3, 5, 23, 14, 14, 131, 233, 84, 42, 664, 2339, 2367, 594, 132, 3166, 18520, 36265, 24714, 4719, 429, 14545, 127511, 408311, 527757, 266219, 40898

Offset: 1

Michel Marcus, Oct 24 2020

See link for the definition of Duck word.

			Triangle begins:
   1;
   2,   3;
   5,  23,   14;
  14, 131,  233,   84;
  42, 664, 2339, 2367, 594;
  ...

Ilani Axelrod-Freed, 312-Avoiding Reduced Valid Hook Configurations and Duck Words, arXiv:2010.11834 [math.CO], 2020. See Definition 5.1 p. 11.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See pp. 15, 17.

Cf. A000108 (column 0), A005700 (diagonal), A005789 (row sums), A031970 (column 1).

cp's OEIS Frontend

A379034 Number of intervals in the lattice of Motzkin paths of length n.

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A379035 Number of intervals in the lattice of Schroeder paths of length 2*n.

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A181571 Third column of triangle in A179898.

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A136440 Sum of heights of all 2-watermelons with wall of length 2*n.

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A336674 Number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+3,n+2,...,1)/(n-1,n-2,...,1).

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A338368 Triangle of number of 312-avoiding reduced valid hook configurations with n hooks that are on permutations with 2n + k points, for n=1 and 1<=k<=n.

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A338403 Regular triangle read by rows: T(n,k) is the number of (n,k)-Duck words, for n>=1 and 0<=k<=n-1.

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