This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A219859 #46 Jan 26 2022 11:11:34
%S A219859 1,1,0,2,2,0,6,18,3,0,24,144,84,4,0,120,1200,1500,300,5,0,720,10800,
%T A219859 23400,10800,930,6,0,5040,105840,352800,294000,63210,2646,7,0,40320,
%U A219859 1128960,5362560,7056000,2857680,324576,7112,8,0,362880,13063680,83825280,160030080,105099120,23496480,1524600,18360,9,0
%N A219859 Triangular array read by rows: T(n,k) is the number of endofunctions, functions f:{1,2,...,n}->{1,2,...,n}, that have exactly k elements with no preimage; n>=0, 0<=k<=n.
%C A219859 Equivalently, T(n,k) is the number of endofunctions whose functional digraph has exactly k leaves.
%C A219859 Equivalently, T(n,k) is the number of doubly rooted trees with k leaves. Here, a doubly rooted tree is a labeled tree in which two special vertices have been selected and the order of the selection matters. [Bona page 266]
%C A219859 Row sums are n^n.
%D A219859 M. Bona, Introduction to Enumerative Combinatorics, McGraw Hill, 2007.
%F A219859 T(n,k) = n!/k! * Stirling2(n,n-k).
%F A219859 T(n,0) = n!.
%F A219859 T(n,k) = A055302(n,k)*(n-k) + A055302(n,k+1)*(k+1). The first term (on rhs of this equation) is the number of such functions in which the preimage of f(n) contains more than one element. The second term is the number of such functions in which the preimage of f(n) contains exactly one element.
%F A219859 T(n,k) = binomial(n,k) Sum_{j=0..n-k}(-1)^j*binomial(n-k,j)*(n-k-j)^n. - _Geoffrey Critzer_, Aug 20 2013
%F A219859 E.g.f.: 1/(1 - (A(x,y) - y*x + x)) where A(x,y) is the e.g.f. for A055302. - _Geoffrey Critzer_, Jan 24 2022
%F A219859 From _Alois P. Heinz_, Jan 24 2022: (Start)
%F A219859 Sum_{k=0..n} k * T(n,k) = A209290(n).
%F A219859 Sum_{k=0..n} (-1)^k * T(n,k) = A344053(n). (End)
%e A219859 Triangle T(n,k) begins:
%e A219859 1;
%e A219859 1, 0;
%e A219859 2, 2, 0;
%e A219859 6, 18, 3, 0;
%e A219859 24, 144, 84, 4, 0;
%e A219859 120, 1200, 1500, 300, 5, 0;
%e A219859 720, 10800, 23400, 10800, 930, 6, 0;
%e A219859 ...
%t A219859 Table[Table[n!/k!StirlingS2[n,n-k],{k,0,n}],{n,0,8}]//Grid
%o A219859 (PARI) row(n) = vector(n+1, k, k--; n!/k! * stirling(n,n-k,2)); \\ _Michel Marcus_, Jan 24 2022
%Y A219859 Column k=0-1 give: A000142, A001804.
%Y A219859 Row sums give A000312.
%Y A219859 T(2n,n) gives A288312.
%Y A219859 Cf. A055302, A055314, A090657, A101817, A209290, A344053.
%K A219859 nonn,tabl
%O A219859 0,4
%A A219859 _Geoffrey Critzer_, Dec 01 2012