A212856 Number of 3 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.

1, 1, 7, 163, 8983, 966751, 179781181, 53090086057, 23402291822743, 14687940716402023, 12645496977257273257, 14490686095184389113277, 21557960797148733086439949, 40776761007750226749220637461, 96332276574683758035941025907591

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..183 (terms n=1..19 from R. H. Hardin)

Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250.

Formula

a(n) = f(n) * n!, where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n+k+1) * f(k) * binomial(n, k)^2 / (n-k)!. - Daniel Suteu, Feb 23 2018

a(n) = (n!)^3 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^3). - Seiichi Manyama, Jul 18 2020

a(n) ~ c * n!^3 / r^n, where r = 1.16151549806386358435938834554462085598002... is the root of the equation HypergeometricPFQ[{}, {1, 1}, -r] = 0 and c = 1.182760720067731330743886867947078139186402925891650811631774628... - Vaclav Kotesovec, Sep 16 2020

Example

Some solutions for n=3:
  2 1 0   2 0 1   1 2 0   0 2 1   2 0 1   2 1 0   2 1 0
  0 2 1   2 0 1   0 2 1   2 1 0   2 1 0   2 1 0   2 0 1
  0 2 1   2 1 0   2 0 1   2 0 1   0 1 2   1 2 0   2 0 1

Maple

A212856 := proc(n) sum(z^k/k!^3, k = 0..infinity);
series(%^x, z=0, n+1): n!^3*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:
seq(A212856(n), n=0..14); # Peter Luschny, May 27 2017
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, -add(
      binomial(n, j)^3*(-1)^j*a(n-j), j=1..n))
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Apr 26 2020

Mathematica

f[0] = 1; f[n_] := f[n] = Sum[(-1)^(n+k+1)*f[k]*Binomial[n, k]^2/(n-k)!, {k, 0, n-1}]; a[n_] := f[n]*n!; Array[a, 14] (* Jean-François Alcover, Feb 27 2018, after Daniel Suteu *)

Crossrefs

Row 3 of A212855.

Cf. A000275, A212857, A212858, A212859, A212860, A336195.

Sequence in context: A364114 A363698 A027549 * A351610 A169608 A184754

Adjacent sequences: A212853 A212854 A212855 * A212857 A212858 A212859

Keyword

nonn

Author

R. H. Hardin, May 28 2012

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 26 2020

Status

approved