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.

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

Offset

0,3

Links

Table of n, a(n) for n=0..12.

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.

Formula

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

Example

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

Mathematica

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

Prog

(PARI) 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

Crossrefs

Cf. A000108.

Cf. A278843, A368019, A368021, A368022, A368023, A368024, A368025.

Column k=0 of A368026.

Sequence in context: A093009 A273442 A322460 * A249787 A264548 A094085

Adjacent sequences: A368009 A368010 A368011 * A368013 A368014 A368015

Keyword

nonn

Author

Stefano Spezia, Dec 08 2023

Status

approved