A355400 -id:A355400 - 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).

A368021 a(n) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+3) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 5, 406, 490614, 8755482505, 2318987094804471, 9179129956137993425772, 546580120389987275414413168012, 492460174883711250780962744103403975159, 6747075036368337341936435881321217868978170152215, 1411689504898999110533224343869931312130954127737962059963934

Offset: 0

Stefano Spezia, Dec 08 2023

nonn

			a(4) = 8755482505:
    5,  14,  42,   132;
   14,  42, 132,   429;
   42, 132, 429,  1430;
  132, 429, 1430, 4862.

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.
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, A000330, A355400.
Cf. A278843, A368012, A368019, A368022, A368023, A368024, A368025.
Column k=3 of A368026.

Join[{1},Table[Permanent[Table[CatalanNumber[i+j+3],{i,0,n-1},{j,0,n-1}]],{n,10}]]

C(n) = binomial(2*n, n)/(n+1); \\ A000108
a(n) = matpermanent(matrix(n, n, i, j, C(i+j+1))); \\ Michel Marcus, Dec 11 2023

Det(M(n)) = A000330(n+1) (see Mays and Wojciechowski, 2000).

A358597 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n+1, such that each p_i is never below p_{i-1}.

Original entry on oeis.org

1, 2, 14, 330, 26026, 6852768, 6018114036, 17618122000050, 171879976152056250, 5586863607659640852000, 604960371578930672694585600, 218201797452928091289631307694720, 262138086905421645845923269465748817136, 1048861003938217198101763464819634006647101600

Offset: 0

Alois P. Heinz, Feb 24 2023

nonn

			a(0) = 1:  ().
                     /\
a(1) = 2:  (/\/\), (/  \).

Alois P. Heinz, Table of n, a(n) for n = 0..61

Cf. A074962, A078920, A123352, A355400.

a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n), i=1..n):
seq(a(n), n=0..14);

Join[{1}, Table[2^(3/2)*n*Gamma[4*n] * BarnesG[n+1] * Sqrt[Gamma[2*n] * BarnesG[4*n]] / BarnesG[3*n + 2], {n, 1, 12}]] (* Vaclav Kotesovec, Aug 26 2023 *)

a(n) = A078920(2n,n) = A123352(2n,n).

a(n) ~ exp(1/24) * 2^(2/3 + 5*n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 + 3*n + 5/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 26 2023

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).

A355503 Total number of m-tuples (p_1, p_2, ..., p_m) of Dyck paths of semilength n-m, such that each p_i is never below p_{i-1} for m=0..n.

Original entry on oeis.org

1, 2, 3, 5, 11, 35, 164, 1120, 10969, 152849, 3029650, 85227078, 3400752392, 192644205130, 15470939367651, 1761760468965521, 284641456742538865, 65175288287611738435, 21159611204475209730138, 9743708333490185603430830, 6357930817596444858142966826

Offset: 0

Alois P. Heinz, Jul 04 2022

nonn

			a(3) = 5: ( ), (/\/\), (//\\), (/\, /\, /\), (<>, <>, <>, <>).

Alois P. Heinz, Table of n, a(n) for n = 0..115
Wikipedia, Counting lattice paths

Cf. A000108, A074962, A078920, A123352, A355400.

Antidiagonal sums of A368025.

a:= n-> add(mul(mul((i+j+2*(n-m))/(i+j), j=i..m-1), i=1..m-1), m=0..n):
seq(a(n), n=0..23);

Table[Sum[Product[Product[(i+j+2*(n-m))/(i+j), {j,i,m-1}], {i,1,m-1}], {m,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)
Table[Sum[BarnesG[1 + m] * Sqrt[BarnesG[1 + 2*n] * BarnesG[2 - 2*m + 2*n] * Gamma[1 + 2*m] * Gamma[1 + n] / (BarnesG[1 + 2*m] * Gamma[1 + m] * Gamma[1 + 2*n] * Gamma[1 - m + n])] / BarnesG[1 - m + 2*n], {m, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2023 *)

a(n) = Sum_{m=0..n} Product_{i=1..m-1, j=i..m-1} (i+j+2*(n-m))/(i+j).
a(n) = 1 + Sum_{k=0..n-1} A078920(n-1,k).
a(n) = 1 + Sum_{k=0..n-1} A123352(n-1,k).
a(n) = Sum_{k=0..n} A368025(n-k, k).
From Vaclav Kotesovec, Aug 27 2023: (Start)
a(n) ~ c * exp(1/24) * 3^(n^2 - n/2) / (sqrt(A) * n^(1/24) * 2^((4*n^2-n-1)/3)), where A = A074962 is the Glaisher-Kinkelin constant and
c = Sum_{k,-oo,oo} 2^((k + mod(n,3)/3)/2 - 3*(k + mod(n,3)/3)^2/2).
Numerically, c = 1.78933741155287907159762028... if mod(n,3)=0 or mod(n,3)=1 and c = 1.78893263307672974352375161... if mod(n,3)=2. (End)

A368298 a(n) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+n) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 1, 53, 490614, 930744290905, 386735380538157813864, 36494318768452684668237864399892, 800075179375382235705309991148469060609055210, 4138855242465150993428071754285859188133806122546895149328625, 5109461743591866972924602083859433690113667142460933537037028649653229023827000

Offset: 0

Stefano Spezia, Dec 20 2023

nonn

			a(3) = 490614:
   5,  14,  42;
  14,  42, 132;
  42, 132, 429.

Wikipedia, Hankel matrix.

Cf. A000108, A355400.

Diagonal of A368025.

with(LinearAlgebra):
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> C(i+j+n-2)))):
seq(a(n), n=0..10);  # Alois P. Heinz, Dec 20 2023

a[n_]:=If[n==0, 1, Permanent[Table[CatalanNumber[i+j+n], {i, 0, n-1}, {j, 0, n-1}]]]; Array[a,10,0]

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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A368021 a(n) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+3) with i,j = 0, ..., n-1.

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A358597 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n+1, such that each p_i is never below p_{i-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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A355503 Total number of m-tuples (p_1, p_2, ..., p_m) of Dyck paths of semilength n-m, such that each p_i is never below p_{i-1} for m=0..n.

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A368298 a(n) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+n) with i,j = 0, ..., n-1.

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Mathematica