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 A214568 #22 Oct 11 2017 05:21:37
%S A214568 1,0,1,0,1,1,0,1,1,2,0,1,2,3,3,0,1,1,6,6,6,0,1,3,7,11,14,12,0,1,1,11,
%T A214568 16,29,32,25,0,1,3,11,26,46,72,75,52,0,1,2,16,27,79,122,182,177,113,0,
%U A214568 1,3,18,42,101,217,336,457,420,247,0,1,1,20,47,149,303,621,911,1160,1005,548
%N A214568 Triangle read by rows: T(n,k) is the number of rooted trees t with n vertices yielding k distinct rooted trees with n+1 vertices when a pendant edge is added to a vertex of t (1 <= k <= n).
%C A214568 Row n contains n entries.
%C A214568 Sum_{k=1..n} T(n,k) = A000081(n) = number of rooted trees with n vertices.
%C A214568 Sum_{k=1..n} k*T(n,k) = A000107(n).
%C A214568 T(n,n) = A004111(n).
%C A214568 T(n,3) = A032741(n-1) = number of proper divisors of n-1; if d is a proper divisor of n-1 (= number of edges), consider d identical rooted trees with (n-1)/d edges, root degree 1, height 2 and identify their roots.
%C A214568 The bivariate g.f. can be computed with eq. (4.2) of Harary-Robinson. - _R. J. Mathar_, Sep 16 2015
%H A214568 Sean A. Irvine, <a href="/A214568/b214568.txt">Rows n=1..44 of triangle, flattened</a>
%H A214568 F. Harary, R. W. Robinson, <a href="http://dx.doi.org/10.1007/BF02579217">Isomorphic factorizations VIII: bisectable trees</a>, Combinatorica 4 (2) (1984) 169-179.
%F A214568 No formula available. Entries have been obtained by counting (using Maple) the rooted trees (identified by their Matula-Goebel numbers) with the required properties (using A061775 and A214567).
%F A214568 Bivariate g.f. T(x,y) = x * y * Product_{p>=1} Product_{k=1..p} (1 + x^p*y^k / (1-x^p))^(a(p,k)), where a(p,k) is the coefficient of x^p*y^k in T(x,y) [(4.2) from Harari and Robinson]. This allows incremental computation of the rows of the sequence by starting with T(x,y) = x*y (p=1) and increasing p by 1 for each row. - _Sean A. Irvine_, Oct 10 2017
%e A214568 Triangle starts:
%e A214568 1;
%e A214568 0, 1;
%e A214568 0, 1, 1;
%e A214568 0, 1, 1, 2;
%e A214568 0, 1, 2, 3, 3;
%e A214568 0, 1, 1, 6, 6, 6;
%e A214568 0, 1, 3, 7, 11, 14, 12;
%e A214568 0, 1, 1, 11, 16, 29, 32, 25;
%e A214568 Row 4 is 0,1,1,2 because the four rooted trees with 4 vertices generate 2,3,4,and 4 rooted trees with 5 vertices.
%Y A214568 Cf. A000081, A000107, A004111, A032741, A061775, A214567.
%K A214568 nonn,tabl
%O A214568 1,10
%A A214568 _Emeric Deutsch_, Jul 26 2012