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.

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

0,4

Links

Table of n, a(n) for n=0..44.

Wikipedia, Hankel matrix.

Example

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, ...
  ...

Maple

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

Mathematica

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

Crossrefs

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).

Sequence in context: A177848 A168549 A010272 * A172093 A294402 A145738

Adjacent sequences: A368023 A368024 A368025 * A368027 A368028 A368029

Keyword

nonn,tabl

Author

Stefano Spezia, Dec 08 2023

Status

approved