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 A144042 #29 Aug 27 2018 23:00:24
%S A144042 1,1,1,1,1,2,1,1,3,4,1,1,4,8,9,1,1,5,13,25,20,1,1,6,19,51,77,48,1,1,7,
%T A144042 26,89,197,258,115,1,1,8,34,141,410,828,871,286,1,1,9,43,209,751,2052,
%U A144042 3526,3049,719,1,1,10,53,295,1260,4337,10440,15538,10834,1842,1,1,11,64
%N A144042 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Euler transform applied k times.
%H A144042 Alois P. Heinz, <a href="/A144042/b144042.txt">Antidiagonals n = 1..141, flattened</a>
%H A144042 M. Bernstein and N. J. A. Sloane, <a href="https://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A144042 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A144042 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%e A144042 Square array begins:
%e A144042 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A144042 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A144042 2, 3, 4, 5, 6, 7, 8, 9, ...
%e A144042 4, 8, 13, 19, 26, 34, 43, 53, ...
%e A144042 9, 25, 51, 89, 141, 209, 295, 401, ...
%e A144042 20, 77, 197, 410, 751, 1260, 1982, 2967, ...
%e A144042 48, 258, 828, 2052, 4337, 8219, 14379, 23659, ...
%e A144042 115, 871, 3526, 10440, 25512, 54677, 106464, 192615, ...
%p A144042 etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
%p A144042 add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
%p A144042 end end:
%p A144042 g:= proc(k) option remember; local b, t; b[0]:= j->
%p A144042 `if`(j<2, j, b[k](j-1)); for t to k do
%p A144042 b[t]:= etr(b[t-1]) od: eval(b[0])
%p A144042 end:
%p A144042 A:= (n, k)-> g(k)(n):
%p A144042 seq(seq(A(n, 1+d-n), n=1..d), d=1..14); # revised _Alois P. Heinz_, Aug 27 2018
%t A144042 etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[Table[A[n, 1 + d-n], {n, 1, d}], {d, 1, 14}] // Flatten (* _Jean-François Alcover_, Dec 20 2013, translated from Maple *)
%Y A144042 Columns k=1-10 give: A000081, A007563, A144035, A144036, A144037, A144038, A144039, A144040, A144041, A305727.
%Y A144042 Rows n=2-4 give: A000012, A000027, A034856.
%Y A144042 Main diagonal gives A305725.
%Y A144042 Cf. A316101.
%K A144042 eigen,nonn,tabl
%O A144042 1,6
%A A144042 _Alois P. Heinz_, Sep 08 2008