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
Keywords
Examples
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
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.
Programs
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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
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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 *)
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
Extensions
a(0)=1 prepended by Alois P. Heinz, Apr 26 2020