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A212850 Number of n X 3 arrays with rows being permutations of 0..2 and no column j greater than column j-1 in all rows.

%I A212850 #40 Sep 08 2019 15:17:02
%S A212850 1,19,163,1135,7291,45199,275563,1666495,10038331,60348079,362442763,
%T A212850 2175719455,13057505371,78354598159,470156286763,2821023814015,
%U A212850 16926401164411,101559181827439,609357415487563,3656151466494175
%N A212850 Number of n X 3 arrays with rows being permutations of 0..2 and no column j greater than column j-1 in all rows.
%C A212850 From _Petros Hadjicostas_, Aug 25 2019: (Start)
%C A212850 Both formulas below follow from the theory in the documentation of array A309951. We have Sum_{s = 0..A000041(3)} (-1)^s * A309951(3,s) * a(n-s) = 0, i.e., a(n) - 10*a(n-1) - 27*a(n-2) + 18*a(n-3) = 0 for n >= 4. This is a consequence of Eq. (6) on p. 248 of Abramson and Promislow (1978), where we let t=0 in the equation.
%C A212850 In the explicit formula by _Vaclav Kotesovec_ below, a(n) = 6^n - 2*3^n + 1^n, the numbers 1, 3, 6 (that are raised to the n-th power) are the multinomial coefficients of the A000041(3) = 3 integer partitions of 3: 1 = 3!/3!, 3 = 3!/(1!2!), 6 = 3!/(1!1!1!).
%C A212850 (End)
%H A212850 R. H. Hardin, <a href="/A212850/b212850.txt">Table of n, a(n) for n = 1..210</a>
%H A212850 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; see Eq. (6) on p. 248 (set t:=0).
%F A212850 Empirical: a(n) = 10*a(n-1) - 27*a(n-2) + 18*a(n-3).
%F A212850 Explicit formula: a(n) = 6^n - 2*3^n + 1. - _Vaclav Kotesovec_, May 31 2012
%e A212850 Some solutions for n=3:
%e A212850   1 2 0   2 1 0   0 2 1   1 2 0   1 2 0   2 1 0   1 2 0
%e A212850   2 0 1   2 0 1   2 0 1   2 0 1   0 2 1   2 0 1   1 0 2
%e A212850   0 2 1   0 1 2   2 1 0   2 1 0   2 0 1   0 2 1   0 2 1
%Y A212850 Column 3 of A212855.
%Y A212850 Cf. A000041, A212851, A212852, A212852, A212853, A212854, A212856, A309951.
%K A212850 nonn
%O A212850 1,2
%A A212850 _R. H. Hardin_, May 28 2012