A065087 -id:A065087 - cp's OEIS frontend

A000313 Number of permutations of length n with 3 consecutive ascending pairs.

0, 0, 0, 1, 4, 30, 220, 1855, 17304, 177996, 2002440, 24474285, 323060540, 4581585866, 69487385604, 1122488536715, 19242660629360, 348933579412440, 6673354706262864, 134252194678935321, 2834212998777523380, 62651024183503148470, 1447238658638922729580

Offset: 1

N. J. A. Sloane

Temporary remark: there may be some issues with respect to the offset of this sequence in the formula and program sections. - Joerg Arndt, Nov 16 2014

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Todd Silvestri, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe)

Cf. A010027, A000255, A000166, A000274, A001260, A001261.

A diagonal in triangle A010027.

series(hypergeom([2,4],[],x/(x+1))/(x+1)^4, x=0, 30); # Mark van Hoeij, Nov 07 2011
a := n -> simplify(hypergeom([4-n,2],[],1))*(-1)^n*(n-1)*(n-2)*(n-3)/6: seq(a(n), n=1..23); # Peter Luschny, Nov 19 2014

Table[(n*(n + 1)!/6)*Sum[(-1)^k/k!, {k, 0, n}], {n, -1, 25}] (* T. D. Noe, Jun 19 2012 *)
a[1]:=0; a[n_Integer/;n>=2]:=(n-2) (n-1) Subfactorial[n-2]/6 (* Todd Silvestri, Nov 15 2014 *)

a = lambda n: (n-2)*(n-1)*sloane.A000166(n-2)/6 if n>2 else 0
[a(n) for n in range(1,24)] # Peter Luschny, Nov 19 2014

a(n) = (n*(n+1)!/6)*sum((-1)^k/k!, k=0..n).
a(n) = A065087(n+2)/3. - Zerinvary Lajos, May 25 2007
E.g.f.: x^3/3!*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
a(n) = round( (exp(-1)*(n+1)!+(-1)^n)*n/6 ). - Mark van Hoeij, Oct 25 2011
G.f.: hypergeom([2, 4],[],x/(x+1))/(x+1)^4. - Mark van Hoeij, Nov 07 2011
a(1) = 0, a(n) = (n-2)*(n-1)*(!(n-2))/6 = (n-2)*(n-1)*A000166(n-2)/6, for n >= 2. - Todd Silvestri, Nov 15 2014
a(n) = hypergeom([4-n,2],[],1)*(-1)^n*A000292(n-3). - Peter Luschny, Nov 19 2014
D-finite with recurrence (-n+4)*a(n) +(n-1)*(n-4)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

More terms from Vladeta Jovovic, Jan 03 2003
Formula added by Sean A. Irvine, Nov 11 2010
Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014

A305730 a(n) is the total displacement of all letters in all permutations of n letters with no fixed points.

Original entry on oeis.org

0, 0, 2, 8, 60, 440, 3710, 34608, 355992, 4004880, 48948570, 646121080, 9163171732, 138974771208, 2244977073430, 38485321258720, 697867158824880, 13346709412525728, 268504389357870642, 5668425997555046760, 125302048367006296940, 2894477317277845459160

Offset: 0

Seiichi Manyama, Jun 22 2018

nonn

			n | 1 2 3 4 | the displacement of all letters | a(n)
--+---------+---------------------------------+------
2 | 2 1     | 1 + 1 = 2                       |   2
3 | 2 3 1   | 1 + 1 + 2 = 4                   |   8
  | 3 1 2   | 2 + 1 + 1 = 4                   |
4 | 2 1 4 3 | 1 + 1 + 1 + 1 = 4               |  60
  | 2 3 4 1 | 1 + 1 + 1 + 3 = 6               |
  | 2 4 1 3 | 1 + 2 + 2 + 1 = 6               |
  | 3 1 4 2 | 2 + 1 + 1 + 2 = 6               |
  | 3 4 1 2 | 2 + 2 + 2 + 2 = 8               |
  | 3 4 2 1 | 2 + 2 + 1 + 3 = 8               |
  | 4 1 2 3 | 3 + 1 + 1 + 1 = 6               |
  | 4 3 1 2 | 3 + 1 + 2 + 2 = 8               |
  | 4 3 2 1 | 3 + 1 + 1 + 3 = 8               |

Seiichi Manyama, Table of n, a(n) for n = 0..400

Cf. A000166, A000313, A065087, A090672.

{a(n) = n*(n+1)!/3*sum(k=0, n, (-1)^k/k!)}

a(n) = n * (n+1) * A000166(n)/3 = 2/3 * A065087(n).
a(n) = n * (n+1)!/3 * Sum_{k=0..n} (-1)^k/k!.
a(n) = n * (n+1) * (a(n-1)/(n-1) + (-1)^n/3) for n > 1.
a(n) = 2 * A000313(n+2). - Alois P. Heinz, Jun 22 2018
E.g.f.: exp(-x)*x^2*(3 - 2*x + x^2)/(3*(1 - x)^3). - Ilya Gutkovskiy, Jun 25 2018

A214261 List of derangements of 1, 2, 3, ..., n for n = 2, 3, 4, ..., in lexicographic order.

Original entry on oeis.org

2, 1, 2, 3, 1, 3, 1, 2, 2, 1, 4, 3, 2, 3, 4, 1, 2, 4, 1, 3, 3, 1, 4, 2, 3, 4, 1, 2, 3, 4, 2, 1, 4, 1, 2, 3, 4, 3, 1, 2, 4, 3, 2, 1, 2, 1, 4, 5, 3, 2, 1, 5, 3, 4, 2, 3, 1, 5, 4, 2, 3, 4, 5, 1, 2, 3, 5, 1, 4, 2, 4, 1, 5, 3, 2, 4, 5, 1, 3, 2, 4, 5, 3, 1, 2, 5, 1

Offset: 2

William Rex Marshall, Jul 08 2012

			The derangements can be written as
21,
231, 312,
2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321, etc.

Cf. A000166, A030298, A065087 (row sums), A086325 (row lengths).

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A000313 Number of permutations of length n with 3 consecutive ascending pairs.

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A305730 a(n) is the total displacement of all letters in all permutations of n letters with no fixed points.

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A214261 List of derangements of 1, 2, 3, ..., n for n = 2, 3, 4, ..., in lexicographic order.

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