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 A034854 #39 Apr 18 2025 22:28:43
%S A034854 3,4,12,5,60,60,6,210,720,360,7,630,6090,7560,2520,8,1736,47040,
%T A034854 112560,80640,20160,9,4536,363384,1496880,1829520,907200,181440,10,
%U A034854 11430,2913120,19207440,36892800,28274400,10886400,1814400
%N A034854 Triangle giving number of labeled trees with n >= 3 nodes and diameter d >= 2.
%H A034854 J. Riordan, <a href="http://www.research.ibm.com/journal/rd/045/ibmrd0405E.pdf">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.[broken link]
%H A034854 J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.
%H A034854 J. Riordan, <a href="/A007401/a007401_8.pdf">The enumeration of trees by height and diameter</a>, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
%H A034854 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A034854 Reference gives recurrence.
%F A034854 From _Geoffrey Critzer_, Aug 02 2022: (Start)
%F A034854 Sum_{d even} a(n,d) = A356292(n) and Sum_{d odd} a(n,d) = A355671(n).
%F A034854 Let G_k(x) be the e.g.f. counting the number of rooted labeled trees with height <= k. Then G_k(x) is defined recursively by G_0(x) = x, G_k(x) = x*exp(G_{k-1}(x)). Let H_k(x) be the e.g.f. counting rooted labeled trees of height k. Then H_0(x) = x, H_k(x) = G_k(x) - G_{k-1}(x) for k >= 1. The e.g.f. for column d = 2*m+1 is H_m(x)^2/2. The e.g.f. for column d = 2*m is G_{m-1}(x)*(exp(H_{m-1}(x)) - 1 - H_{m-1}(x)). (End)
%e A034854 Triangle begins:
%e A034854 3;
%e A034854 4, 12;
%e A034854 5, 60, 60;
%e A034854 6, 210, 720, 360;
%e A034854 7, 630, 6090, 7560, 2520;
%e A034854 ...
%Y A034854 Cf. A001852, A034855, A356292, A355671.
%K A034854 nonn,tabl,easy,nice
%O A034854 0,1
%A A034854 _N. J. A. Sloane_
%E A034854 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004
%E A034854 Name corrected by _Geoffrey Critzer_, Aug 02 2022