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A206823 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n.

%I A206823 #32 Feb 07 2020 20:26:51
%S A206823 1,0,1,2,0,2,3,18,0,6,40,48,144,0,24,205,1000,600,1200,0,120,2556,
%T A206823 7380,18000,7200,10800,0,720,24409,125244,180810,294000,88200,105840,
%U A206823 0,5040,347712,1562176,4007808,3857280,4704000,1128960,1128960,0,40320
%N A206823 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n.
%C A206823 Row sums = n^n, all functions f:{1,2,...,n}->{1,2,...,n}.
%C A206823 T(n,n)= n!, bijections on {1,2,...,n}.
%H A206823 Alois P. Heinz, <a href="/A206823/b206823.txt">Table of n, a(n) for n = 0..140, flattened</a>
%F A206823 E.g.f.: Sum_{k=0..n} T(n,k) * y^k * x^n / n!  = (exp(x) - x + y*x)^n.
%e A206823 Triangle T(n,k) begins:
%e A206823     1;
%e A206823     0      1;
%e A206823     2      0     2;
%e A206823     3     18     0      6;
%e A206823    40     48   144      0    24;
%e A206823   205   1000   600   1200     0     120;
%e A206823   ...
%p A206823 with(combinat): C:= binomial:
%p A206823 b:= proc(t, i, u) option remember; `if`(t=0, 1,
%p A206823       `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
%p A206823       *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
%p A206823     end:
%p A206823 T:= (n, k)-> C(n, k)*C(n, k)*k! *b(n-k$2, n-k):
%p A206823 seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Nov 13 2013
%t A206823 nn = 8; Prepend[CoefficientList[Table[n! Coefficient[Series[(Exp[x] - x + y x)^n, {x, 0, nn}], x^n], {n, 1, nn}], y], {1}] // Flatten
%Y A206823 Row sums give: A000312.
%Y A206823 Column k=0 gives: A231797.
%Y A206823 Cf. A231602.
%K A206823 nonn,tabl
%O A206823 0,4
%A A206823 _Geoffrey Critzer_, Feb 12 2012