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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. 8
1, 1, 31, 7291, 7225951, 21855093751, 164481310134301, 2675558106868421881, 84853928323286139485791, 4849446032811641059203617551, 469353176282647626764795665676281, 73159514984813223626195834388445570381, 17619138865526260905773841471696025142373661 (list; graph; refs; listen; history; text; internal format)
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

CROSSREFS

Row 5 of A212855.

Cf. A000012, A000225, A000275, A212850, A212851, A212852, A212853, A212854, A212856, A212857, A212859, A212860, A336197.

Sequence in context: A065756 A263378 A306840 * A059384 A136676 A135811

Adjacent sequences:  A212855 A212856 A212857 * A212859 A212860 A212861

KEYWORD

nonn

AUTHOR

R. H. Hardin, May 28 2012

EXTENSIONS

a(0)=1 prepended by Seiichi Manyama, Jul 18 2020

STATUS

approved

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Last modified August 5 21:50 EDT 2020. Contains 336213 sequences. (Running on oeis4.)