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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.
9
1, 1, 31, 7291, 7225951, 21855093751, 164481310134301, 2675558106868421881, 84853928323286139485791, 4849446032811641059203617551, 469353176282647626764795665676281, 73159514984813223626195834388445570381, 17619138865526260905773841471696025142373661
OFFSET
0,3
COMMENTS
From Petros Hadjicostas, Sep 08 2019: (Start)
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).
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.
(End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..120 (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; see Eq. (8) on p. 249.
FORMULA
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
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
EXAMPLE
Some solutions for n=3:
2 0 1 0 1 2 0 2 1 0 2 1 1 2 0 0 2 1 2 0 1
2 0 1 2 1 0 0 1 2 0 2 1 0 1 2 1 2 0 2 0 1
0 1 2 2 0 1 0 2 1 2 1 0 0 1 2 0 1 2 2 1 0
2 0 1 0 1 2 1 2 0 0 2 1 1 0 2 2 1 0 1 0 2
1 2 0 0 2 1 2 1 0 1 2 0 0 1 2 2 1 0 2 1 0
MAPLE
A212858 := proc(n) sum(z^k/k!^5, k = 0..infinity);
series(%^x, z=0, n+1): n!^5*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:
seq(A212858(n), n=1..12); # Peter Luschny, May 27 2017
MATHEMATICA
T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k - j], {j, 1, k}]];
a[n_] := T[5, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz in A212855 *)
KEYWORD
nonn
AUTHOR
R. H. Hardin, May 28 2012
EXTENSIONS
a(0)=1 prepended by Seiichi Manyama, Jul 18 2020
STATUS
approved