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 A058127 #48 Feb 17 2022 07:56:08
%S A058127 1,1,1,1,2,3,1,3,8,16,1,4,15,50,125,1,5,24,108,432,1296,1,6,35,196,
%T A058127 1029,4802,16807,1,7,48,320,2048,12288,65536,262144,1,8,63,486,3645,
%U A058127 26244,177147,1062882,4782969,1,9,80,700,6000,50000,400000,3000000,20000000,100000000
%N A058127 Triangle read by rows: T(j,k) is the number of acyclic functions from {1,...,j} to {1,...,k}. For n >= 1, a(n) = (k-j)*k^(j-1), where k is such that C(k,2) < n <= C(k+1,2) and j = (n-1) mod C(k,2). Alternatively, table T(k,j) read by antidiagonals with k >= 1, 0 <= j <= k: T(k,j) = number of acyclic-function digraphs on k vertices with j vertices of outdegree 1 and (k-j) vertices of outdegree 0; T(k,j) = (k-j)*k^(j-1).
%C A058127 An acyclic function f from domain D={1,...,j} to codomain C={1,...,k} is a function such that, for every subset A of D, f(A) does not equal A. Equivalently, an acyclic function f "eventually sends" under successive composition all elements of D to {j+1,...,k}. An acyclic-function digraph G is a labeled directed graph that satisfies (i) all vertices have outdegree 0 or 1; (ii) if vertex x has outdegree 0 and vertex y has outdegree 1, then x > y; (iii) G has no cycles and no loops. There is a one-to-one correspondence between acyclic functions from D to C and acyclic-function digraphs with j vertices of outdegree 1 and j-k vertices of outdegree 0.
%C A058127 n-th row of the triangle is the n-th iterate of "perform binomial transform operation" (bto) on current row to get next row, extracting the leftmost n terms for n-th row (i.e., all terms left of the zero). First row is (bto): [1, -1, 0, 0, 0, ...]. 5th row is 1, 4, 15, 50, 125, since (bto) performed 5 times iteratively on [1, -1, 0, 0, 0, ...] = 1, 4, 15, 50, 125, 0, -31, ... - _Gary W. Adamson_, Apr 30 2005
%C A058127 T(k,j) can be shown to be equal to the number of spanning trees of the complete graph on k vertices that contain a specific subtree with k-j-1 edges. - _John L. Chiarelli_, Oct 04 2016
%C A058127 T(k-1, j-1) is also the number of parking functions with j cars and k spots (see Theorem 2.2 in Kenyon and Yin). - _Stefano Spezia_, Apr 09 2021
%H A058127 T. D. Noe, <a href="/A058127/b058127.txt">Rows n = 1..50 of triangle, flattened</a>
%H A058127 Richard Kenyon and Mei Yin, <a href="https://arxiv.org/abs/2103.17180">Parking functions: From combinatorics to probability</a>, arXiv:2103.17180 [math.CO] (2021).
%H A058127 Henri Mühle, <a href="http://fpsac2019.fmf.uni-lj.si/resources/Proceedings/29.pdf">Ballot-Noncrossing Partitions</a>, Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana), Séminaire Lotharingien de Combinatoire (2019) Vol. 82B, Article #7.
%H A058127 Jim Pitman and Richard P. Stanley, <a href="https://doi.org/10.1007/s00454-002-2776-6">A polytope related to empirical distributions, plane trees, parking functions, and the associahedron</a>, Discrete Comput. Geom. 27: 603-634 (2002).
%H A058127 D. P. Walsh, <a href="http://www.mtsu.edu/~dwalsh/acyclic/acycnote.html">Notes on acyclic functions and their directed graphs</a>
%F A058127 For fixed m = k-j, a(n) = T(k, j) = T(m+j, j) = m*(m+j)^(j-1). Exponential generating function g for T(m+j, j) = m*(m+j)^(j-1) is given by g(t) = exp(-m*W(-t)), where W denotes the principal branch of Lambert's W function. Lambert's W function satisfies W(t)*exp(W(t)) = t for t >= -exp(-1).
%F A058127 T(n, k) = Sum_{i=0..k} T(n-1, i) * binomial(k, i) if k < n. - _Michael Somos_, Sep 20 2017
%e A058127 a(6) = T(3,2) = 3 because there are 3 acyclic functions from {1,2} to {1,2,3}: {(1,2),(2,3)}, {(1,3),(2,3)} and {(1,3),(2,1)}.
%e A058127 Triangle begins:
%e A058127 1;
%e A058127 1, 1;
%e A058127 1, 2, 3;
%e A058127 1, 3, 8, 16;
%e A058127 1, 4, 15, 50, 125;
%e A058127 1, 5, 24, 108, 432, 1296;
%e A058127 1, 6, 35, 196, 1029, 4802, 16807;
%e A058127 1, 7, 48, 320, 2048, 12288, 65536, 262144;
%e A058127 ...
%p A058127 T := proc(n,k) (n-k)*n^(k-1) end; seq(print(seq(T(n,k),k=0..n-1)),n=1..9); # _Peter Luschny_, Jan 14 2009
%t A058127 t[n_, k_] := (n-k)*n^(k-1); Table[t[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Dec 03 2013 *)
%o A058127 (Magma) /* As triangle */ [[(n-k)*n^(k-1): k in [0..n-1]]: n in [1.. 10]]; // _Vincenzo Librandi_, Aug 11 2017
%o A058127 (PARI) {T(n, k) = if( k<0 || k>n, 0, n==0, 1, (n-k) * n^(k-1))}; /* _Michael Somos_, Sep 20 2017 */
%Y A058127 The sum of antidiagonals is A058128. The sequence b(n) = T(n, n-1) for n >= 1 is A000272, labeled trees on n nodes.
%Y A058127 The sequence c(n) = T(n, n-2) for n >= 2 is A007334(n). The sequence d(n) = T(n, n-3) for n >= 3 is A089463(n-3,0). - _Peter Luschny_, Apr 22 2009
%K A058127 nice,nonn,tabl
%O A058127 1,5
%A A058127 _Dennis P. Walsh_, Nov 14 2000
%E A058127 a(32) corrected by _T. D. Noe_, Jan 25 2008