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A212858 Number of 5 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.

%I A212858 #42 Apr 01 2024 10:10:03
%S A212858 1,1,31,7291,7225951,21855093751,164481310134301,2675558106868421881,
%T A212858 84853928323286139485791,4849446032811641059203617551,
%U A212858 469353176282647626764795665676281,73159514984813223626195834388445570381,17619138865526260905773841471696025142373661
%N A212858 Number of 5 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.
%C A212858 From _Petros Hadjicostas_, Sep 08 2019: (Start)
%C A212858 We generalize _Daniel Suteu_'s recurrence from A212856. Notice first that, in the notation of Abramson and Promislow (1978), we have a(n) = R(m=5, n, t=0).
%C A212858 Letting y=0 in Eq. (8), p. 249, of Abramson and Promislow (1978), we get 1 + Sum_{n >= 1} R(m,n,t=0)*x^n/(n!)^m = 1/f(-x), where f(x) = Sum_{i >= 0} (x^i/(i!)^m). Matching coefficients, we get Sum_{s = 1..n} R(m, s, t=0) * (-1)^(s-1) * binomial(n,s)^m = 1, from which the recurrence in the Formula section follows.
%C A212858 (End)
%H A212858 Seiichi Manyama, <a href="/A212858/b212858.txt">Table of n, a(n) for n = 0..120</a> (terms n=1..19 from R. H. Hardin)
%H A212858 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. (8) on p. 249.
%F A212858 a(n) = (-1)^(n-1) + Sum_{s = 1..n-1} a(s) * (-1)^(n-s-1) * binomial(n,s)^m for n >= 2 with a(1) = 1. Here m = 5. - _Petros Hadjicostas_, Sep 08 2019
%F A212858 a(n) = (n!)^5 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^5). (see Petros Hadjicostas's comment on Sep 08 2019) - _Seiichi Manyama_, Jul 18 2020
%e A212858 Some solutions for n=3:
%e A212858   2 0 1   0 1 2   0 2 1   0 2 1   1 2 0   0 2 1   2 0 1
%e A212858   2 0 1   2 1 0   0 1 2   0 2 1   0 1 2   1 2 0   2 0 1
%e A212858   0 1 2   2 0 1   0 2 1   2 1 0   0 1 2   0 1 2   2 1 0
%e A212858   2 0 1   0 1 2   1 2 0   0 2 1   1 0 2   2 1 0   1 0 2
%e A212858   1 2 0   0 2 1   2 1 0   1 2 0   0 1 2   2 1 0   2 1 0
%p A212858 A212858 := proc(n) sum(z^k/k!^5, k = 0..infinity);
%p A212858 series(%^x, z=0, n+1): n!^5*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
%p A212858 seq(A212858(n), n=1..12); # _Peter Luschny_, May 27 2017
%t A212858 T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k - j], {j, 1, k}]];
%t A212858 a[n_] := T[5, n];
%t A212858 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Apr 01 2024, after _Alois P. Heinz_ in A212855 *)
%Y A212858 Row 5 of A212855.
%Y A212858 Cf. A000012, A000225, A000275, A212850, A212851, A212852, A212853, A212854, A212856, A212857, A212859, A212860, A336197.
%K A212858 nonn
%O A212858 0,3
%A A212858 _R. H. Hardin_, May 28 2012
%E A212858 a(0)=1 prepended by _Seiichi Manyama_, Jul 18 2020