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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).

%I A225942 #27 May 22 2013 19:14:56
%S A225942 1,1,2,2,6,21,24,192,40,120,1800,1205,720,18000,25680,2256,5040,
%T A225942 194040,489510,134953,40320,2257920,9031680,5196800,250496,362880,
%U A225942 28304640,167015520,166793760,24943689,3628800,381024000,3149798400,4904524800,1514960640,46063360
%N 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).
%C A225942 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.
%C A225942 Row sums = n^n.
%C A225942 T(n,0)   = n!.
%C A225942 T(2n,n)  = A209289(n).
%H A225942 Alois P. Heinz, <a href="/A225942/b225942.txt">Rows n = 0..100, flattened</a>
%F A225942 T(n,k) = n! * [x^n*y^(2k)] (y*cosh(x)+sinh(x))^n.
%e A225942 1;
%e A225942 1;
%e A225942 2,    2;
%e A225942 6,    21;
%e A225942 24,   192,    40;
%e A225942 120,  1800,   1205;
%e A225942 720,  18000,  25680,  2256;
%e A225942 5040, 194040, 489510, 134953;
%t A225942 Map[Select[#, # > 0 &] &, Prepend[Table[nn = n;
%t A225942     CoefficientList[
%t A225942      Expand[n! Coefficient[
%t A225942         Series[(y Cosh[x] + Sinh[x])^n, {x, 0, nn}], x^n]], y], {n, 1,
%t A225942       7}], {1}]] // Grid
%K A225942 nonn,tabf
%O A225942 0,3
%A A225942 _Geoffrey Critzer_, May 21 2013