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 A319254 #22 Apr 20 2023 14:56:52
%S A319254 1,2,1,3,3,2,4,6,10,5,5,10,28,40,12,6,15,60,156,170,33,7,21,110,430,
%T A319254 948,785,90,8,28,182,965,3396,6206,3770,261,9,36,280,1890,9376,28818,
%U A319254 42504,18805,766,10,45,408,3360,21798,97775,256172,301548,96180,2312
%N A319254 Array read by antidiagonals: T(n,k) is the number of series-reduced rooted trees with n leaves of k colors.
%C A319254 Not all colors need to be used.
%C A319254 See table 2.3 in the Johnson reference.
%H A319254 Andrew Howroyd, <a href="/A319254/b319254.txt">Table of n, a(n) for n = 1..1275</a>
%H A319254 Virginia Perkins Johnson, <a href="https://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. South Carolina, 2012.
%e A319254 Array begins:
%e A319254 ==================================================================
%e A319254 n\k| 1 2 3 4 5 6 7
%e A319254 ---+--------------------------------------------------------------
%e A319254 1 | 1 2 3 4 5 6 7 ...
%e A319254 2 | 1 3 6 10 15 21 28 ...
%e A319254 3 | 2 10 28 60 110 182 280 ...
%e A319254 4 | 5 40 156 430 965 1890 3360 ...
%e A319254 5 | 12 170 948 3396 9376 21798 44856 ...
%e A319254 6 | 33 785 6206 28818 97775 269675 642124 ...
%e A319254 7 | 90 3770 42504 256172 1068450 3496326 9632960 ...
%e A319254 8 | 261 18805 301548 2357138 12081605 46897359 149491104 ...
%e A319254 9 | 766 96180 2195100 22253672 140160650 645338444 2379859608 ...
%e A319254 ...
%p A319254 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A319254 add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
%p A319254 end:
%p A319254 A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
%p A319254 seq(seq(A(n, 1+d-n), n=1..d), d=1..12); # _Alois P. Heinz_, Sep 17 2018
%t A319254 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
%t A319254 A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
%t A319254 Table[A[n, 1 + d - n], {d, 1, 12}, {n, 1, d}] // Flatten (* _Jean-François Alcover_, Sep 11 2019, after _Alois P. Heinz_ *)
%o A319254 (PARI) \\ here R(n,k) gives k'th column as a vector.
%o A319254 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A319254 R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v,[0]))[n])); v}
%o A319254 {my(T=Mat(vector(8, k, R(8, k)~))); for(n=1, #T~, print(T[n,]))} \\ _Andrew Howroyd_, Sep 15 2018
%Y A319254 Columns 1..5 are A000669, A050381, A220823, A220824, A220825.
%Y A319254 Main diagonal is A319369.
%Y A319254 Cf. A141610, A242249, A255517, A256064, A256068, A319376.
%K A319254 nonn,tabl
%O A319254 1,2
%A A319254 _Andrew Howroyd_, Sep 15 2018