A368023 -id:A368023 - cp's OEIS frontend

A278843 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = Catalan(i+j).

1, 2, 53, 19148, 97432285, 7146659536022, 7683122105385590481, 122557371932066196769721048, 29280740446653388021872592300048913, 105552099397122165176384278493772205485181002, 5775235099464970103806328103231969172586171168151193533

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

			From _Stefano Spezia_, Dec 08 2023: (Start)
a(4) = 97432285:
   2,   5,  14,   42;
   5,  14,  42,  132;
  14,  42, 132,  429;
  42, 132, 429, 1430.
(End)

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, A277829, A278770, A278844.
Cf. A368012, A368019, A368021, A368022, A368023, A368024, A368025.
Column k=2 of A368026.

Flatten[{1, Table[Permanent[Table[CatalanNumber[i+j], {i, 1, n}, {j, 1, n}]], {n, 1, 14}]}]

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

Det(M(n)) = n + 1 (see Mays and Wojciechowski, 2000). - Stefano Spezia, Dec 08 2023

A368026 Array read by ascending antidiagonals: A(n, k) 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+k) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 95, 9, 2, 1, 38057, 979, 53, 5, 1, 207372681, 1417675, 19148, 406, 14, 1, 15977248385955, 28665184527, 97432285, 490614, 3612, 42, 1, 17828166968924572623, 8325587326635565, 7146659536022, 8755482505, 14798454, 35442, 132, 1, 292842668371666277607183121, 35389363346700690999467, 7683122105385590481, 2318987094804471, 930744290905, 499114473, 372801, 429, 1

Offset: 0

Stefano Spezia, Dec 08 2023

			The array begins:
      1,       1,        1,          1,            1, ...
      1,       1,        2,          5,           14, ...
      3,       9,       53,        406,         3612, ...
     95,     979,    19148,     490614,     14798454, ...
  38057, 1417675, 97432285, 8755482505, 930744290905, ...
  ...

Wikipedia, Hankel matrix.

Cf. A000012 (n=0), A000108 (n=1).
Cf. A368012 (k=0), A368019 (k=1), A278843 (k=2), A368021 (k=3), A368022 (k=4), A368023 (k=5), A368024 (k=6).
Cf. A368025 (determinant), A368298 (diagonal).

with(LinearAlgebra):
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= (n, k)-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> C(i+j+k-2)))):
seq(seq(A(d-k, k), k=0..d), d=0..8);  # Alois P. Heinz, Dec 20 2023

A[n_,k_]:=If[n==0,1,Permanent[Table[CatalanNumber[i+j+k],{i,0,n-1},{j,0,n-1}]]]; Table[A[n-k,k],{n,0,8},{k,0,n}]//Flatten

A368012 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) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 1, 3, 95, 38057, 207372681, 15977248385955, 17828166968924572623, 292842668371666277607183121, 71645110588632775032727941092738473, 263399284865064400938403105805219201386749363, 14653009564320804036813733761485114583670416021283903839, 12403293423772370760211339634714413308535752478944832963336911564521

Offset: 0

Stefano Spezia, Dec 08 2023

nonn

			a(4) = 38057:
  1,  1,  2,   5;
  1,  2,  5,  14;
  2,  5, 14,  42;
  5, 14, 42, 132.

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.
Cf. A278843, A368019, A368021, A368022, A368023, A368024, A368025.
Column k=0 of A368026.

Join[{1},Table[Permanent[Table[CatalanNumber[i+j],{i,0,n-1},{j,0,n-1}]],{n,12}]]

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

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

A368019 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+1) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 1, 9, 979, 1417675, 28665184527, 8325587326635565, 35389363346700690999467, 2230867495754739989535874468003, 2106171270085074740753132799048111935155, 30007898337707083458776293190436074888346472515407, 6491219550166075876771081259839537013093735814742318424677245

Offset: 0

Stefano Spezia, Dec 08 2023

nonn

			a(4) = 1417675:
   1,  2,   5,  14;
   2,  5,  14,  42;
   5, 14,  42, 132;
  14, 42, 132, 429.

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.
Cf. A278843, A368012, A368021, A368022, A368023, A368024, A368025.
Column k=1 of A368026.

Join[{1},Table[Permanent[Table[CatalanNumber[i+j+1],{i,0,n-1},{j,0,n-1}]],{n,11}]]

C(n) = binomial(2*n,n)/(n+1); \\ A000108
a(n) = matpermanent(matrix(n,n,i,j,C(i+j-1))); \\ Michel Marcus, Dec 09 2023

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

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).

A368022 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+4) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 14, 3612, 14798454, 930744290905, 891107801867703108, 12977575456694246217097712, 2880177942851157900010279504962852, 9767068920318918290905853040035029840419305, 507521146153330160633968276251306280235282377512091202, 405202219609475677155580649938116235991326716758748940659564085180

Offset: 0

Stefano Spezia, Dec 08 2023

nonn

			a(4) = 930744290905:
   14,   42,  132,   429;
   42,  132,  429,  1430;
  132,  429, 1430,  4862;
  429, 1430, 4862, 16796.

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, A006858.
Cf. A278843, A368012, A368019, A368021, A368023, A368024, A368025.
Column k=4 of A368026.

Join[{1},Table[Permanent[Table[CatalanNumber[i+j+4],{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+2))); \\ Michel Marcus, Dec 11 2023

Det(M(n)) = A006858(n+1).

A368024 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+6) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 132, 372801, 18271508684, 14570336513383508, 184204867131613485842464, 36494318768452684668237864399892, 112700882376631374264115400599310944646268, 5412697889621813132124427516447652973723355158580585, 4039897382110972290799421201399595435416108353911344509968785100

Offset: 0

Stefano Spezia, Dec 08 2023

nonn

			a(4) = 14570336513383508:
   132,   429,  1430,   4862;
   429,  1430,  4862,  16796;
  1430,  4862, 16796,  58786;
  4862, 16796, 58786, 208012.

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, A335857 (determinant).
Cf. A278843, A368012, A368019, A368021, A368022, A368023, A368025.
Column k=6 of A368026.

Join[{1},Table[Permanent[Table[CatalanNumber[i+j+6],{i,0,n-1},{j,0,n-1}]],{n,9}]]

C(n) = binomial(2*n, n)/(n+1); \\ A000108
a(n) = matpermanent(matrix(n, n, i, j, C(i+j+4))); \\ Michel Marcus, Dec 11 2023

cp's OEIS Frontend

A278843 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = Catalan(i+j).

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A368026 Array read by ascending antidiagonals: A(n, k) 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+k) with i,j = 0, ..., n-1.

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A368012 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) with i,j = 0, ..., n-1.

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A368019 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+1) 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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A368022 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+4) with i,j = 0, ..., n-1.

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A368024 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+6) with i,j = 0, ..., n-1.

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