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 A034940 #31 Aug 30 2018 22:40:48
%S A034940 1,3,75,5145,688905,152193195,50174679555,23089081640625,
%T A034940 14140034726843025,11119632520038117075,10920803043967635894075,
%U A034940 13100477280449146440878025,18849023772776126861572265625,32038907667175368299033846026875,63516199119599233704934379969701875
%N A034940 Number of rooted labeled triangular cacti with 2n+1 nodes (n triangles).
%D A034940 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307. (4.2.44)
%H A034940 Alois P. Heinz, <a href="/A034940/b034940.txt">Table of n, a(n) for n = 0..100</a>
%H A034940 Maryam Bahrani and Jérémie Lumbroso, <a href="http://arxiv.org/abs/1608.01465">Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition</a>, arXiv:1608.01465 [math.CO], 2016.
%H A034940 Wikipedia, <a href="http://en.wikipedia.org/wiki/Series_reversion">Lagrange Inversion Theorem</a>
%H A034940 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%F A034940 a(n) = b(2*n+1) where e.g.f. of b satisfies B(x)=x*exp(B(x)^2/2).
%F A034940 The closed form a(n) = (2n-1)!! (2n+1)^n can be obtained from the generating function. - _Noam D. Elkies_, Dec 16 2002
%F A034940 From Peter Bala, Jul 31 2012: (Start)
%F A034940 E.g.f. A(x) = series reversion of x*exp(-1/2*x^2) = sum {n >= 0} a(n)*x^(2*n+1)/(2*n+1)! = x + 3*x^3/3! + 75*x^5/5! + .... The Lagrange inversion formula gives a(n) = (2*n+1)^n*(2*n)!/(2^n*n!).
%F A034940 A(x)^2 = T(x^2), where T denotes the tree function T(x) := sum {n >= 1} n^(n-1)*x^n/n!. A(x)^r = sum {n >= 0} r*(2*n+r)^(n-1)*x^(2*n+r)/(2^n*n!).
%F A034940 x = A(x)*exp(-1/2*A(x)^2). dA/dx = exp(1/2*A^2)/(1-A^2).
%F A034940 Let the function F(x) = A(exp(x)). Then dF/dx = F/(1-F^2). More generally, (d/dx)^(n+1)(F) is a rational function in F(x) given by (d/dx)^(n+1)(F) = F*R(n,F^2)/(1-F^2)^(2*n+1), where R(n,x) is the n-th row generating polynomial of A214406.
%F A034940 (End)
%e A034940 E.g. a(3) = 5!! 7^3 = (1*3*5) * 343 = 5145.
%e A034940 From _Peter Bala_, Jul 31 2012: (Start)
%e A034940 Relation with rows of A214406: F(x) := A(exp(x)).
%e A034940 (d/dx)^1(F) = F/(1-F^2)
%e A034940 (d/dx)^2(F)) = F*(1 + F^2)/(1 - F^2)^3
%e A034940 (d/dx)^3(F)) = F*(1 + 8*F^2 + 3*F^4)/(1 - F^2)^5
%e A034940 (d/dx)^4(F)) = F*(1 + 33*F^2 + 71*F^4 + 15*F^6)/(1 - F^2)^7
%e A034940 (End)
%t A034940 a[n_] := (2*n-1)!!*(2*n+1)^n; Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, May 13 2013, after _Noam D. Elkies_ *)
%o A034940 (PARI) a(n) = (2*n+1)^n*(2*n)!/(2^n*n!); \\ _Andrew Howroyd_, Aug 30 2018
%Y A034940 Cf. A003080, A000169, A214406.
%K A034940 nonn,eigen
%O A034940 0,2
%A A034940 _Christian G. Bower_, Oct 15 1998
%E A034940 a(10) corrected by _Jean-François Alcover_, May 13 2013
%E A034940 a(12)-a(14) from _Alois P. Heinz_, Jul 08 2015