A185025 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).

1, 1, 3, 1, 18, 9, 163, 90, 3, 1950, 1100, 75, 28821, 16245, 1575, 15, 505876, 283122, 33810, 735, 10270569, 5699932, 780150, 26460, 105, 236644092, 130267440, 19615932, 884520, 8505, 6098971555, 3332614725, 538325550, 29619450, 467775, 945

Offset: 0

Author

Geoffrey Critzer, Dec 24 2012

Keywords

• nonn
• tabf

Comments

It appears that as n gets large, row n conforms to a Poisson distribution with mean = 1/2. In other words, as n gets large, T(n,k) approaches n^n/(2^k*k!*e^(1/2)).

Examples

			Triangle begins:
           1;
           1;
           3,          1;
          18,          9;
         163,         90,         3;
        1950,       1100,        75;
       28821,      16245,      1575,       15;
      505876,     283122,     33810,      735;
    10270569,    5699932,    780150,    26460,    105;
   236644092,  130267440,  19615932,   884520,   8505;
  6098971555, 3332614725, 538325550, 29619450, 467775, 945;
  ...
		

Crossrefs

Column k=0 gives A089466.

Cf. A000169, A081131.

Programs

Mathematica

nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}]; Range[0,nn]! CoefficientList[Series[Exp[t^2/2(y-1)]/(1-t), {x,0,nn}], {x,y}]//Grid

Formula

E.g.f.: exp((T(x)^2/2)*(y-1))/(1 - T(x)) where T(x) is the e.g.f. for A000169.

Sum_{k=1..floor(n/2)} k * T(n,k) = A081131(n).