cp's OEIS Frontend

A228601 Triangle read by rows: T(n,k) is the number of trees with n vertices and having k distinct rootings (1 <= k <= n).

%I A228601 #23 Oct 11 2017 21:47:53
%S A228601 1,1,0,0,1,0,0,2,0,0,0,1,1,1,0,0,2,1,2,1,0,0,1,2,4,1,2,1,0,2,1,7,4,4,
%T A228601 4,1,0,1,2,7,7,9,10,8,3,0,2,3,12,10,17,19,20,17,6,0,1,2,12,14,28,37,
%U A228601 45,46,35,15,0,2,1,18,21,46,60,87,106,103,78,29
%N A228601 Triangle read by rows: T(n,k) is the number of trees with n vertices and having k distinct rootings (1 <= k <= n).
%C A228601 The entries in the triangle have been obtained - painstakingly - from the Read & Wilson reference (pp. 63-73); the white vertices indicate the possible distinct rootings for the given tree.
%D A228601 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%H A228601 Sean A. Irvine, <a href="/A228601/b228601.txt">Rows n=1..44 of triangle, flattened</a>
%H A228601 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 A228601 Sum of entries in row n = A000055(n).
%F A228601 Sum_{k=1..n} k*T(n,k)  = A000081(n).
%F A228601 T(n,n) = A000220(n).
%F A228601 Let A214568(x,y) be the bivariate g.f. of A214568, then this g.f. is A214568(x,y) -( [A214568(x,y)]^2 + A214568(x^2,y^2) )/2 + A214568(x^2,y), see eq. (4.8) by Harary-Robinson. - _R. J. Mathar_, Sep 16 2015
%e A228601 Row 4 is 0,2,0,0 because the trees with 4 vertices are (i) the path tree abcd with 2 distinct rootings (at a and at b) and (ii) the star tree with 4 vertices having, obviously, 2 distinct rootings.
%e A228601 Triangle starts:
%e A228601   1;
%e A228601   1, 0;
%e A228601   0, 1, 0;
%e A228601   0, 2, 0, 0;
%e A228601   0, 1, 1, 1, 0;
%e A228601   0, 2, 1, 2, 1, 0;
%e A228601   0, 1, 2, 4, 1, 2, 1;
%Y A228601 Cf. A000055, A000081, A000220.
%K A228601 nonn,tabl,look
%O A228601 1,8
%A A228601 _Emeric Deutsch_, Oct 20 2013