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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.

%I A212856 #39 Sep 16 2020 05:52:21
%S A212856 1,1,7,163,8983,966751,179781181,53090086057,23402291822743,
%T A212856 14687940716402023,12645496977257273257,14490686095184389113277,
%U A212856 21557960797148733086439949,40776761007750226749220637461,96332276574683758035941025907591
%N 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.
%H A212856 Alois P. Heinz, <a href="/A212856/b212856.txt">Table of n, a(n) for n = 0..183</a> (terms n=1..19 from R. H. Hardin)
%H A212856 Morton Abramson and David Promislow, <a href="https://doi.org/10.1016/0097-3165(78)90012-2">Enumeration of arrays by column rises</a>, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250.
%F A212856 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
%F A212856 a(n) = (n!)^3 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^3). - _Seiichi Manyama_, Jul 18 2020
%F A212856 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
%e A212856 Some solutions for n=3:
%e A212856   2 1 0   2 0 1   1 2 0   0 2 1   2 0 1   2 1 0   2 1 0
%e A212856   0 2 1   2 0 1   0 2 1   2 1 0   2 1 0   2 1 0   2 0 1
%e A212856   0 2 1   2 1 0   2 0 1   2 0 1   0 1 2   1 2 0   2 0 1
%p A212856 A212856 := proc(n) sum(z^k/k!^3, k = 0..infinity);
%p A212856 series(%^x, z=0, n+1): n!^3*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
%p A212856 seq(A212856(n), n=0..14); # _Peter Luschny_, May 27 2017
%p A212856 # second Maple program:
%p A212856 a:= proc(n) option remember; `if`(n=0, 1, -add(
%p A212856       binomial(n, j)^3*(-1)^j*a(n-j), j=1..n))
%p A212856     end:
%p A212856 seq(a(n), n=0..15);  # _Alois P. Heinz_, Apr 26 2020
%t A212856 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_ *)
%Y A212856 Row 3 of A212855.
%Y A212856 Cf. A000275, A212857, A212858, A212859, A212860, A336195.
%K A212856 nonn
%O A212856 0,3
%A A212856 _R. H. Hardin_, May 28 2012
%E A212856 a(0)=1 prepended by _Alois P. Heinz_, Apr 26 2020