A378112 -id:A378112 - cp's OEIS frontend

A368025 Array read by ascending antidiagonals: A(n,k) is the determinant of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+k) with i,j = 0, ..., n-1.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 14, 14, 1, 1, 1, 5, 30, 84, 42, 1, 1, 1, 6, 55, 330, 594, 132, 1, 1, 1, 7, 91, 1001, 4719, 4719, 429, 1, 1, 1, 8, 140, 2548, 26026, 81796, 40898, 1430, 1, 1, 1, 9, 204, 5712, 111384, 884884, 1643356, 379236, 4862, 1

Offset: 0

Stefano Spezia, Dec 08 2023

This array is a variant of the triangles A078920 and A123352 extended to the trivial cases (here for k=0).

			The array begins:
  1, 1, 1,   1,    1,      1,        1, ...
  1, 1, 2,   5,   14,     42,      132, ...
  1, 1, 3,  14,   84,    594,     4719, ...
  1, 1, 4,  30,  330,   4719,    81796, ...
  1, 1, 5,  55, 1001,  26026,   884884, ...
  1, 1, 6,  91, 2548, 111384,  6852768, ...
  1, 1, 7, 140, 5712, 395352, 41314284, ...
  ...

Arthur T. Benjamin, Naiomi T. Cameron, Jennifer J. Quinn, and Carl R. Yerger, Catalan determinants-a combinatorial approach, Congressus Numerantium 200, 27-34 (2010). On ResearchGate.
Jishe Feng, The explicit formula of Hankel determinant with Catalan elements, arXiv:2010.06586 [math.GM], 2020.
M. E. Mays and Jerzy Wojciechowski, A determinant property of Catalan numbers. Discrete Math. 211, No. 1-3, 125-133 (2000).
Wikipedia, Hankel matrix.

Cf. A000108 (n=1), A005700 (n=2), A006149 (n=3), A006150 (n=4), A006151 (n=5).
Cf. A000012 (k=0 or k=1 or n=0), A000330, A078920, A091962, A123352, A335857 (k=6).
Cf. A355400, A368026 (permanent), A378112.
Antidiagonal sums give A355503.

A:= proc(n, k) option remember; `if`(k=0, 1, 2^n*mul(
      (2*(k-i)+2*n-3)/(k+2*n-1-i), i=0..n-1)*A(n, k-1))
    end:
seq(seq(A(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Dec 20 2023

A[n_,k_]:=If[n==0,1,Det[Table[CatalanNumber[i+j+k],{i,0,n-1},{j,0,n-1}]]]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

For an explicit formula of A(n,k), see equation (5) in Feng, 2020.
A(n,2) = n + 1.
A(n,3) = A000330(n+1).
A(n,4) = A006858(n+1).
A(n,5) = A091962(n+1).
Diagonal: A(n,n) = A123352(2*n-1,n-1) = A355400(n).

A378113 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_n only touches the x-axis at its endpoints.

Original entry on oeis.org

1, 1, 2, 23, 880, 105554, 40446551, 50637232553, 209584899607676, 2881189188022646406, 131778113962930341491415, 20065327661524165382215337625, 10173706896856510992170168595911888, 17178054578218938036671513200907244799852, 96590987238453485101729361602126273065518820938

Offset: 0

Alois P. Heinz, Nov 16 2024

nonn

			a(2) = 2:
          /\       /\   /\
   (/\/\,/  \)   (/  \,/  \) .
The a(3) = 23 3-tuples can be encoded as 114, 115, 124, 125, 134, 135, 144, 145, 155, 224, 225, 244, 245, 255, 334, 335, 344, 345, 355, 444, 445, 455, 555, where the digits represent the following Dyck paths:
  1        2        3        4        5 /\
            /\         /\     /\/\     /  \
  /\/\/\   /  \/\   /\/  \   /    \   /    \ .

Alois P. Heinz, Table of n, a(n) for n = 0..62
Wikipedia, Counting lattice paths

Main diagonal of A378112.

Cf. A000108, A355400.

b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(
      (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))
    end:
A:= proc(n, k) option remember;
      b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..15);

a(n) = A378112(n,n).

A355281 Number of pairs of nested Dyck paths from (0,0) to (n,n) such that the upper path only touches the diagonal at its endpoints.

Original entry on oeis.org

1, 1, 2, 9, 55, 400, 3266, 28999, 274537, 2734885, 28401315, 305352146, 3380956839, 38394091370, 445702108969, 5274935433915, 63507021523471, 776347636736261, 9621502184089320, 120726786082609207, 1531938384684090884, 19639252409244653785, 254143269904958943103, 3317204158078663935592

Offset: 0

Joel B. Lewis, Jun 26 2022

nonn

Let B be the 2 X n X n box of integer points with opposite corners (0, 0, 0) and (1, n - 1, n - 1). For n >= 1, a(n) is also the number of plane partitions that fit inside B and whose cells lie on or below the plane x + y + z = n - 1. Proof: after rotating by 90 degrees, the upper Dyck path is the outer boundary of the region of the plane partition filled with 2's and the lower Dyck path is the outer boundary of the region of the plane partition filled with 1's or 2's.

Alois P. Heinz, Table of n, a(n) for n = 0..841

Cf. A000108, A005700.

Column k=2 of A378112.

b:= proc(n) option remember; `if`(n=0, 1,
      b(n-1)*((4*n)^2-4)/(n+2)/(n+3))
    end:
a:= proc(n) option remember;
      b(n)-add(a(n-i)*b(i), i=1..n-1)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 26 2022

nmax = 23;
c = CatalanNumber;
B[x_] = Sum[(c[n] c[n+2] - c[n+1]^2) x^n, {n, 0, nmax}];
CoefficientList[2 - 1/B[x] + O[x]^(nmax+1), x] (* Jean-François Alcover, Jul 06 2022 *)

G.f.: 2 - 1/B(x) where B(x) is the generating function for A005700.

INVERTi transform of A005700.

A378114 Number of 3-tuples (p_1, p_2, p_3) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_3 only touches the x-axis at its endpoints.

Original entry on oeis.org

1, 1, 3, 23, 265, 3942, 70395, 1445700, 33188889, 834702890, 22656163450, 656075013591, 20085981787831, 645418018740113, 21637970282382744, 753157297564682541, 27105935164769925549, 1005184072184843625837, 38295251586474334236780, 1495061191885030011433707

Offset: 0

Alois P. Heinz, Nov 16 2024

nonn

			a(2) = 3:
               /\            /\   /\       /\   /\   /\
   (/\/\,/\/\,/  \)   (/\/\,/  \,/  \)   (/  \,/  \,/  \)  .
The a(3) = 23 3-tuples can be encoded as 114, 115, 124, 125, 134, 135, 144, 145, 155, 224, 225, 244, 245, 255, 334, 335, 344, 345, 355, 444, 445, 455, 555, where the digits represent the following Dyck paths:
  1        2        3        4        5 /\
            /\         /\     /\/\     /  \
  /\/\/\   /  \/\   /\/  \   /    \   /    \ .

Alois P. Heinz, Table of n, a(n) for n = 0..567
Wikipedia, Counting lattice paths

Column k=3 of A378112.

Cf. A000108, A006149.

b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(
      (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))
    end:
A:= proc(n, k) option remember;
      b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)
    end:
a:= n-> A(n, 3):
seq(a(n), n=0..20);

INVERTi transform of A006149.

cp's OEIS Frontend

A368025 Array read by ascending antidiagonals: A(n,k) is the determinant of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+k) with i,j = 0, ..., n-1.

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A378113 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_n only touches the x-axis at its endpoints.

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A355281 Number of pairs of nested Dyck paths from (0,0) to (n,n) such that the upper path only touches the diagonal at its endpoints.

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A378114 Number of 3-tuples (p_1, p_2, p_3) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_3 only touches the x-axis at its endpoints.

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