A034940 Number of rooted labeled triangular cacti with 2n+1 nodes (n triangles).

1, 3, 75, 5145, 688905, 152193195, 50174679555, 23089081640625, 14140034726843025, 11119632520038117075, 10920803043967635894075, 13100477280449146440878025, 18849023772776126861572265625, 32038907667175368299033846026875, 63516199119599233704934379969701875

Offset

0,2

References

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 307. (4.2.44)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.

Wikipedia, Lagrange Inversion Theorem

Index entries for sequences related to cacti

Formula

a(n) = b(2*n+1) where e.g.f. of b satisfies B(x)=x*exp(B(x)^2/2).

The closed form a(n) = (2n-1)!! (2n+1)^n can be obtained from the generating function. - Noam D. Elkies, Dec 16 2002

From Peter Bala, Jul 31 2012: (Start)

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!).

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!).

x = A(x)*exp(-1/2*A(x)^2). dA/dx = exp(1/2*A^2)/(1-A^2).

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.

(End)

Example

E.g. a(3) = 5!! 7^3 = (1*3*5) * 343 = 5145.
From Peter Bala, Jul 31 2012: (Start)
Relation with rows of A214406: F(x) := A(exp(x)).
(d/dx)^1(F) = F/(1-F^2)
(d/dx)^2(F)) = F*(1 + F^2)/(1 - F^2)^3
(d/dx)^3(F)) = F*(1 + 8*F^2 + 3*F^4)/(1 - F^2)^5
(d/dx)^4(F)) = F*(1 + 33*F^2 + 71*F^4 + 15*F^6)/(1 - F^2)^7
(End)

Mathematica

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 *)

Prog

(PARI) a(n) = (2*n+1)^n*(2*n)!/(2^n*n!); \\ Andrew Howroyd, Aug 30 2018

Crossrefs

Cf. A003080, A000169, A214406.

Sequence in context: A361537 A249938 A195263 * A300924 A183290 A336062

Adjacent sequences: A034937 A034938 A034939 * A034941 A034942 A034943

Keyword

nonn,eigen

Author

Christian G. Bower, Oct 15 1998

Extensions

a(10) corrected by Jean-François Alcover, May 13 2013

a(12)-a(14) from Alois P. Heinz, Jul 08 2015

Status

approved