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 A187105 #16 Jul 06 2022 04:30:21
%S A187105 1,1,1,3,2,1,10,8,3,1,41,38,15,4,1,196,216,90,24,5,1,1057,1402,633,
%T A187105 172,35,6,1,6322,10156,5028,1424,290,48,7,1,41393,80838,44217,13204,
%U A187105 2745,450,63,8,1,293608,698704,424434,134680,28900,4776,658,80,9,1
%N A187105 Triangle T(n,k) read by rows: number of height-2-restricted finite functions.
%C A187105 Triangle T(n,k) with 1 <= k <= n+1 is the number of functions f:[n+1-k]->[n+1] such that f(f(f(x))) is undefined, that is, either f(x) or f(f(x)) is in {n+2-k,...,n+1}. Such functions are called height-2 restricted functions. Note that the null function, which occurs when k=n+1, vacuously satisfies the conditions for a height-2 restricted function, and hence T(n,n+1)=1. The sequence a(n)=T(n,1) is sequence A000248, the number of forests with n nodes and height at most 1. The height of a function f:D->C, with D a proper subset of finite C, is the maximum h such that (f^h)(x) exists for some x in D. A height restricted function f is acyclic since, if x is in a cycle of f, then (f^z)(x) exists for all positive integers z. [Note that [m] denotes the set of the first m positive integers and that f^m denotes the m-fold self-composition of f so that (f^0)(x)=x, (f^1)(x)=f(x),(f^2)(x)=f(f(x)), etc.]
%H A187105 Dennis Walsh, <a href="http://frank.mtsu.edu/~dwalsh/HEIGHT2.pdf"> Notes on height-restricted finite functions</a>
%F A187105 T(n,k) = Sum_{j=0..n+1-k}binomial(n+1-k,j)*k^j*j^(n+1-k-j) for n>=0 and T(0,k) for k>=1.
%F A187105 E.g.f. of column k: exp(k*x*exp(x)).
%F A187105 With t(n,k) = T(n+k-1,k), t(n,k+j) = Sum_{i=0..n}binomial(n,i)*t(i,k)*t(n-i,j).
%e A187105 Triangle of initial terms:
%e A187105 1
%e A187105 1 1
%e A187105 3 2 1
%e A187105 10 8 3 1
%e A187105 41 38 15 4 1
%e A187105 196 216 90 24 5 1
%e A187105 1057 1402 633 172 35 6 1
%e A187105 T(4,3) = 15 since there are 15 functions f:[2]->[5] such that either f(x) or f(f(x)) is in {3,4,5}. Using <f(1),f(2)> to denote these functions we have the following 15 functions: <2,3>, <2,4>, <2,5>, <3,1>, <3,3>, <3,4>, <3,5>, <4,1>, <4,3>, <4,4>, <4,5>, <5,1>, <5,3>, <5,4>, <5,5>.
%p A187105 seq(seq(sum(binomial(n+1-k,j)*k^j*j^(n+1-k-j),j=0..(n+1-k)),k=1..n),n=1..15); # triangle's right edge of ones is omitted with this program
%t A187105 t[n_, k_] := If[ k == n + 1, 1, Sum[ Binomial[n + 1 - k, j]*k^j*j^(n + 1 - k - j), {j, 0, n + 1 - k}]]; Table[ t[n, k], {n, 0, 9}, {k, n + 1}] // Flatten
%Y A187105 Cf. A000248, A275707, A355501.
%K A187105 nonn,nice,easy,tabl
%O A187105 1,4
%A A187105 _Dennis P. Walsh_, Mar 04 2011