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A212860 Number of 7 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, 127, 275563, 4479288703, 347190069843751, 96426023622482278621, 78785944892341703819175577, 163925632052722656731213188429183, 777880066963402408939826643081996101263, 7717574897043522397037273525233635595811018377 (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=7, 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..92 (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 = 7. - Petros Hadjicostas, Sep 08 2019

a(n) = (n!)^7 * [x^n] 1 / (1 + Sum_{k>=1} (-x)^k / (k!)^7). (see Petros Hadjicostas's comment on Sep 08 2019) - Seiichi Manyama, Jul 18 2020

EXAMPLE

Some solutions for n=3:

  0 1 2   0 1 2   0 2 1   0 1 2   0 2 1   0 2 1   0 2 1

  1 2 0   0 2 1   0 2 1   1 0 2   0 2 1   1 0 2   2 1 0

  1 0 2   2 1 0   2 0 1   0 1 2   2 0 1   1 0 2   1 2 0

  0 2 1   1 0 2   0 2 1   1 0 2   0 1 2   2 0 1   0 1 2

  2 0 1   2 1 0   1 0 2   2 1 0   1 2 0   0 1 2   1 2 0

  2 1 0   0 1 2   1 0 2   0 1 2   2 0 1   1 0 2   2 1 0

  1 2 0   2 1 0   0 1 2   0 2 1   2 1 0   2 0 1   2 0 1

MAPLE

A212860 := proc(n) sum(z^k/k!^7, k = 0..infinity);

series(%^x, z=0, n+1): n!^7*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:

seq(A212860(n), n=1..10); # Peter Luschny, May 27 2017

CROSSREFS

Row 7 of A212855.

Cf. A000012, A000225, A000275, A212850, A212851, A212852, A212853, A212854, A212856, A212857, A212858, A212859.

Sequence in context: A203798 A195218 A215692 * A334668 A135813 A112016

Adjacent sequences:  A212857 A212858 A212859 * A212861 A212862 A212863

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 June 15 15:16 EDT 2021. Contains 345049 sequences. (Running on oeis4.)