A225942 Triangular array read by rows: T(n,k) is the number of f:{1,2,...,n}->{1,2,...,n} with exactly 2k elements that have a preimage of even (possibly zero) cardinality; n>=0, 0<=k<=floor(n/2).

1, 1, 2, 2, 6, 21, 24, 192, 40, 120, 1800, 1205, 720, 18000, 25680, 2256, 5040, 194040, 489510, 134953, 40320, 2257920, 9031680, 5196800, 250496, 362880, 28304640, 167015520, 166793760, 24943689, 3628800, 381024000, 3149798400, 4904524800, 1514960640, 46063360

Offset: 0

Author

Geoffrey Critzer, May 21 2013

Keywords

• nonn
• tabf

Comments

Urn A is initially filled with n labeled balls while urn B is empty. A ball is randomly selected and switched from one urn to the other. T(n,k)/n^n is the probability that urn A contains 2k balls after n switches have been made.

Row sums = n^n.

T(n,0) = n!.

T(2n,n) = A209289(n).

Examples

			1;
1;
2,    2;
6,    21;
24,   192,    40;
120,  1800,   1205;
720,  18000,  25680,  2256;
5040, 194040, 489510, 134953;
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened

Programs

Mathematica

Map[Select[#, # > 0 &] &, Prepend[Table[nn = n;
    CoefficientList[
     Expand[n! Coefficient[
        Series[(y Cosh[x] + Sinh[x])^n, {x, 0, nn}], x^n]], y], {n, 1,
      7}], {1}]] // Grid

Formula

T(n,k) = n! * [x^n*y^(2k)] (y*cosh(x)+sinh(x))^n.