A277829 -id:A277829 - cp's OEIS frontend

A278770 Second series of Hankel determinants based on squares of Catalan numbers.

1, 4, 159, 81296, 585396881, 61994262028020, 98925461617709743975, 2414583243140269424293854400, 910504281815476426073145299359052745, 5341354769384557074743892800174971438265757284, 489946515248844365403775650233194419858267427195735348151, 705379807799940807283682167156246485805791300481296966713394135535056

Offset: 0

Karol A. Penson, Nov 28 2016

nonn

It would be useful to know the formula for this sequence.

Cf. A000108, A277829.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (t-> (binomial(2*t, t)/(t+1))^2)(i+j))):
seq(a(n), n=0..12);  # Alois P. Heinz, May 01 2018

Flatten[{1, Table[Det[Table[(CatalanNumber[i + j])^2, {i, n}, {j, n}]], {n, 11}]}]

Conjecture: lim n->infinity log(a(n))/n^2 = 2*log(2). - Vaclav Kotesovec, Nov 28 2016

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

Original entry on oeis.org

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

A278860 First series of Hankel determinants based on hyperfactorial.

Original entry on oeis.org

1, 1, 92, 7207016256, 22448940392028699561050505216, 462177945344267713413229252637478222244311831261385379020800000

Offset: 0

Karol A. Penson, Nov 29 2016

nonn

It would be interesting to know the formula for this sequence.

Cf. A002109, A277829, A278770.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (t-> mul(k^k, k=0..t))(i+j-1))):
seq(a(n), n=0..6);  # Alois P. Heinz, Nov 29 2016

Table[Det[Table[Hyperfactorial[i + j - 1], {i, n}, {j, n}]], {n, 7}]

A278868 Second series of Hankel determinants based on hyperfactorial/4.

Original entry on oeis.org

1, 1, 6183, 5772211367657472, 76148812142946816440318638031477145600000, 3940613226283843476344831941863494501303228636304800836707599745608602091520000000000

Offset: 0

Karol A. Penson, Nov 29 2016

nonn

It would be useful to know the formula for this sequence.

Cf. A002109, A277829, A278770, A278860.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (t-> mul(k^k, k=0..t)/4)(i+j))):
seq(a(n), n=0..6);  # Alois P. Heinz, Nov 29 2016

Table[Det[Table[Hyperfactorial[i + j]/4, {i, n}, {j, n}]], {n, 6}]

A278844 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = (Catalan(i+j))^2.

Original entry on oeis.org

1, 4, 1409, 61813936, 405546824579185, 444429790193462299152820, 87560380592182813232163111416947497, 3267701159348966740482726536189228228570094452288, 24017165813923278639658993746183865352216465064857235206429739929

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Cf. A000108, A277829, A278770, A278843.

Flatten[{1, Table[Permanent[Table[CatalanNumber[i+j]^2, {i, 1, n}, {j, 1, n}]], {n, 1, 12}]}]

A278897 First series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

Original entry on oeis.org

1, 1, 14, 146275425484, 558429168112511379835233509679413804180016

Offset: 0

Karol A. Penson, Nov 30 2016

nonn

If we regard Bell(k^2) as the k-th Stieltjes moment for k>=0, then the solution of the Stieltjes moment problem is given in the P. Blasiak et al. reference, see below. We conjecture that a(n)>0 for n>=0. The positivity of these Hankel determinants a(n), n>=0 is one of the conditions for the existence of a positive solution. Apparently this solution is not unique.

P. Blasiak, K. A. Penson and A. I. Solomon, Dobinsky-type relations and the log-normal distribution, J. Phys. A: Math. Gen. 36, L273 (2003), arXiv: quant-ph/0303030, 2003.

Cf. A000110, A277829, A278770, A278868, A278860.

with(LinearAlgebra), with(combinat):
h20:=(i,j)->bell((i+j-2)^2):
seq(Determinant(Matrix(kk,kk,h20)),kk=0..6);

Table[Det[Table[BellB[(i + j - 2)^2], {i, n}, {j, n}]], {n, 6}], n=>1.

A278903 Second series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

Original entry on oeis.org

1, 1, 20922, 96938760190744854628604, 1039473181175725249030299777705981025900981837012416973957739853576960

Offset: 0

Karol A. Penson, Nov 30 2016

nonn

If we regard Bell(k^2) as the k-th Stieltjes moment for k>=0, then the solution of the Stieltjes moment problem is given in the P. Blasiak et al. reference, see below. We conjecture that a(n)>0 for n>=0. The positivity of these Hankel determinants a(n), n>=0 is one of the conditions for the existence of a positive solution. Apparently this solution is not unique.

P. Blasiak, K. A. Penson and A. I. Solomon, Dobinsky-type relations and the log-normal distribution, J. Phys. A: Math. Gen. 36, L273 (2003), arXiv: quant-ph/0303030, 2003.

Cf. A000110, A277829, A278770, A278868, A278860, A278897.

with(LinearAlgebra), with(combinat):
h21:=(i, j)->bell((i+j-1)^2):
seq(Determinant(Matrix(kk, kk, h21)), kk=0..6);

Table[Det[Table[BellB[(i + j - 1)^2], {i, n}, {j, n}]], {n, 5}], n=>1.

A303826 Hankel transform of A001246.

Original entry on oeis.org

1, 3, 99, 43881, 280974025, 26916213134875, 39339805703866118875, 887919033897631593738548625, 311967217568836709207331125906048625, 1715750319988362944273302140220635494624999475

Offset: 0

Karol A. Penson, May 01 2018

nonn

a(n) is the determinant of the (n+1) X (n+1) matrix A defined by A[i,j] = A001246(i+j-2) for 1 <= i,j <= n+1. - Altug Alkan, May 01 2018

			a(2) = 99 because determinant of the following matrix is 99.
[1  1   4]
[1  4  25]
[4 25 196]

Vaclav Kotesovec, Table of n, a(n) for n = 0..41
Wikipedia, Hankel matrix

Cf. A000108, A001246, A277829, A278770.

a:= n-> LinearAlgebra[Determinant](Matrix(n+1, (i, j)->
        (t-> (binomial(2*t, t)/(t+1))^2)(i+j-2))):
seq(a(n), n=0..15);  # Alois P. Heinz, May 01 2018

Table[Det[
  Table[(CatalanNumber[i + j - 2])^2, {i, 1, n + 1}, {j, 1, n + 1}]], {n,
  0, 10}]

a(n) = matdet(matrix(n+1, n+1, i, j, (binomial(2*(i+j-2),(i+j-2))/(i+j-1))^2)); \\ Altug Alkan, May 01 2018

a(n)^(1/n) ~ d * 4^n, where d = 0.263565... - Vaclav Kotesovec, May 06 2024

Name simplified by Andrey Zabolotskiy, May 02 2018

cp's OEIS Frontend

A278770 Second series of Hankel determinants based on squares of Catalan numbers.

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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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A278860 First series of Hankel determinants based on hyperfactorial.

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A278868 Second series of Hankel determinants based on hyperfactorial/4.

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A278844 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = (Catalan(i+j))^2.

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Mathematica

A278897 First series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

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Maple

Mathematica

A278903 Second series of Hankel determinants based on Bell numbers of argument k^2, Bell(k^2).

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Maple

Mathematica

A303826 Hankel transform of A001246.

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