cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A008291 Triangle of rencontres numbers.

Original entry on oeis.org

1, 2, 3, 9, 8, 6, 44, 45, 20, 10, 265, 264, 135, 40, 15, 1854, 1855, 924, 315, 70, 21, 14833, 14832, 7420, 2464, 630, 112, 28, 133496, 133497, 66744, 22260, 5544, 1134, 168, 36, 1334961, 1334960, 667485, 222480, 55650, 11088, 1890, 240, 45, 14684570
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Comments

T(n,k) = number of permutations of n elements with k fixed points.
T(n,n-1)=0 and T(n,n)=1 are omitted from the array. - Geoffrey Critzer, Nov 28 2011.

Examples

			Triangle begins:
       1
       2      3
       9      8     6
      44     45    20    10
     265    264   135    40   15
    1854   1855   924   315   70   21
   14833  14832  7420  2464  630  112  28
  133496 133497 66744 22260 5544 1134 168 36
  ...

References

  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.
  • Kaufmann, Arnold. "Introduction a la combinatorique en vue des applications." Dunod, Paris, 1968. See p. 92.
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Links

Crossrefs

Cf. A008290, A170942.
Diagonals give A000217, A007290, A060008, A060836, A000166, A000240, A000387, A000449, A000475.
Row sums give A033312.
Cf. A320582.

Programs

  • Maple
    T:= proc(n, k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
      (T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))
    end:
seq(seq(T(n, k), k=0..n-2), n=2..12);  # Alois P. Heinz, Mar 17 2013
  • Mathematica
    Prepend[Flatten[f[list_]:=Select[list,#>1&];Map[f,Drop[Transpose[Table[d = Exp[-x]/(1 - x);Range[0, 10]! CoefficientList[Series[d x^k/k!, {x, 0, 10}],x], {k, 0, 8}]], 3]]], 1] (* Geoffrey Critzer, Nov 28 2011 *)
  • PARI
    T(n, k)= if(k<0 || k>n, 0, n!/k!*sum(i=0, n-k, (-1)^i/i!))

Formula

T(n,k) = binomial(n,k)*A000166(n-k) = A008290(n,k).
E.g.f. for column k: (x^k/k!)(exp(-x)/(1-x)). - Geoffrey Critzer, Nov 28 2011
Row generating polynomials appear to be given by -1 + sum {k = 0..n} (-1)^(n+k)*C(n,k)*(1+k*x)^(n-k)*(2+(k-1)*x)^k. - Peter Bala, Dec 29 2011

Extensions

Comments and more terms from Michael Somos, Apr 26 2000

A060008 a(n) = 9*binomial(n,4) = 3n*(n-1)*(n-2)*(n-3)/8.

Original entry on oeis.org

0, 0, 0, 0, 9, 45, 135, 315, 630, 1134, 1890, 2970, 4455, 6435, 9009, 12285, 16380, 21420, 27540, 34884, 43605, 53865, 65835, 79695, 95634, 113850, 134550, 157950, 184275, 213759, 246645, 283185, 323640, 368280, 417384, 471240, 530145, 594405
Offset: 0

Views

Author

Henry Bottomley, Mar 16 2001

Keywords

Comments

Number of permutations of n letters where exactly four change position.

Examples

			a(6) = 135 since there are 15 ways to choose the four points that move and 9 ways to move them and 15*9 = 135.

Links

Crossrefs

For changing 0, 1, 2, 3, 4, 5, n-4, n elements see A000012, A000004, A000217 (offset), A007290, A060008, A060836, A000475, A000166. Also see A000332, A008290.
A diagonal of A008291.

Programs

  • Mathematica
    9*Binomial[Range[0,40],4] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,0,0,0,9},40] (* Harvey P. Dale, Jun 09 2014 *)
  • PARI
    a(n) = { 3*n*(n - 1)*(n - 2)*(n - 3)/8 } \\ Harry J. Smith, Jul 01 2009

Formula

Equals 3*A050534. - Zerinvary Lajos, Feb 12 2007
G.f.: 9*x^4/(1-x)^5. - Colin Barker, Jul 02 2012
From Amiram Eldar, Jul 19 2022: (Start)
Sum_{n>=4} 1/a(n) = 4/27.
Sum_{n>=4} (-1)^n/a(n) = 32*log(2)/9 - 64/27. (End)
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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