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A090657 Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).

%I A090657 #51 Jan 02 2022 18:29:10
%S A090657 1,0,1,0,2,2,0,3,18,6,0,4,84,144,24,0,5,300,1500,1200,120,0,6,930,
%T A090657 10800,23400,10800,720,0,7,2646,63210,294000,352800,105840,5040,0,8,
%U A090657 7112,324576,2857680,7056000,5362560,1128960,40320
%N A090657 Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).
%C A090657 Another version is in A101817. - _Philippe Deléham_, Feb 16 2013
%H A090657 Alois P. Heinz, <a href="/A090657/b090657.txt">Rows n = 0..62, flattened</a>
%H A090657 C. M. Ringel, <a href="http://arxiv.org/abs/1502.06553">The Catalan combinatorics of the hereditary artin algebras</a>, arXiv preprint arXiv:1502.06553, 2015
%F A090657 T(n,k) = C(n,k) * k! * A048993(n,k).
%F A090657 T(n,k) = A008279(n,k) * A048993(n,k).
%F A090657 T(n,k) = C(n,k) * A019538(n, k).
%F A090657 T(n,k) = C(n,k) * Sum_{j=0..k} (-1)^(k-j) * C(k,j) * j^n.
%F A090657 T(n,k) = n * (T(n-1,k-1) + k/(n-k) * T(n-1,k)) with T(n,n) = n! and T(n,0) = 0 for n>0.
%F A090657 T(2n,n) = A288312(n). - _Alois P. Heinz_, Jun 07 2017
%e A090657 Triangle begins:
%e A090657   1;
%e A090657   0,  1;
%e A090657   0,  2,   2;
%e A090657   0,  3,  18,   6;
%e A090657   0,  4,  84, 144, 24;
%e A090657   ...
%p A090657 T:= proc(n,k) option remember;
%p A090657       if k=n then n!
%p A090657     elif k=0 or k>n then 0
%p A090657     else n * (T(n-1,k-1) + k/(n-k) * T(n-1,k))
%p A090657       fi
%p A090657     end:
%p A090657 seq(seq(T(n,k), k=0..n), n=0..10);
%t A090657 Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 0, n}], {n, 0,10}] // Flatten  (* _Geoffrey Critzer_, Sep 09 2011 *)
%Y A090657 Row sums give: A000312. Columns k=0-2 give: A000007, A001477, A068605. Diagonal, lower diagonal give: A000142, A001804. Cf. A007318, A048993, A019538, A008279.
%Y A090657 Cf. A101817, A288312.
%K A090657 easy,nonn,tabl
%O A090657 0,5
%A A090657 _Philippe Deléham_, Dec 14 2003
%E A090657 Revised description from Jan Maciak, Apr 25 2004
%E A090657 Edited by _Alois P. Heinz_, Jan 17 2011