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%I #25 Oct 04 2018 20:12:33
%S 1,1,1,1,1,0,1,1,1,0,1,1,2,1,0,1,1,3,3,1,0,1,1,4,6,5,1,0,1,1,5,10,14,
%T 7,1,0,1,1,6,15,30,27,11,1,0,1,1,7,21,55,75,58,15,1,0,1,1,8,28,91,170,
%U 206,111,22,1,0,1,1,9,36,140,336,571,518,223,30,1,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.
%C A(n,k) is the number of unlabeled rooted trees with exactly n leaves, all in level k. A(3,3) = 6:
%C : o o o o o o
%C : | | | / \ / \ /|\
%C : o o o o o o o o o o
%C : | / \ /|\ | | ( ) | | | |
%C : o o o o o o o o o o o o o o
%C : /|\ ( ) | | | | ( ) | | | | | | |
%C : o o o o o o o o o o o o o o o o o o
%H Alois P. Heinz, <a href="/A290353/b290353.txt">Antidiagonals n = 0..140, flattened</a>
%H B. A. Huberman and T. Hogg, <a href="https://doi.org/10.1016/0167-2789(86)90308-1">Complexity and adaptation</a>, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F G.f. of column k=0: 1+x, of column k>0: Product_{j>0} 1/(1-x^j)^A(j,k-1).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
%e 0, 1, 3, 6, 10, 15, 21, 28, 36, ...
%e 0, 1, 5, 14, 30, 55, 91, 140, 204, ...
%e 0, 1, 7, 27, 75, 170, 336, 602, 1002, ...
%e 0, 1, 11, 58, 206, 571, 1337, 2772, 5244, ...
%e 0, 1, 15, 111, 518, 1789, 5026, 12166, 26328, ...
%e 0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, ...
%p with(numtheory):
%p A:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
%p add(A(d, k-1)*d, d=divisors(j))*A(n-j, k), j=1..n)/n))
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[A[d, k - 1]*d, {d, Divisors[j]}] A[n - j, k], {j, n}]/n]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}]//Flatten (* _Indranil Ghosh_, Jul 30 2017, after Maple code *)
%Y Columns k=1-10 give: A000012, A000041, A001970, A007713, A007714, A290355, A290356, A290357, A290358, A290359.
%Y Rows 0+1,2-10 give: A000012, A001477, A000217, A000330, A007715, A290360, A290361, A290362, A290363, A290364.
%Y Main diagonal gives A290354.
%Y Cf. A144150.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Jul 28 2017