A049130 Revert transform of ((x - 1)(x + 1))/(-1 - x + x^3).

1, 1, 2, 4, 10, 26, 73, 211, 630, 1918, 5944, 18668, 59311, 190243, 615269, 2004025, 6568174, 21645438, 71681152, 238416580, 796107464, 2667768904, 8968626418, 30240087086, 102238147891, 346514952331, 1177137322768, 4007326361986, 13669068510355, 46711170248183, 159899495303170

Offset

1,3

Comments

Binomial transform of aerated version of A001764. - Paul Barry, Nov 05 2006

a(n) is the number of planar rooted trees with n vertices, where each vertex has one or an even number of children. - Kurusch Ebrahimi-Fard and Dominique Manchon (kef(AT)unizar.es), Jun 05 2010

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1808

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

K. Ebrahimi-Fard and D. Manchon, Dendriform Equations, Journal of Algebra, 322 (2009), 4053-4079.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Index entries for reversions of series

Formula

a(n) = sum{k=0..floor(n/2), C(n,2k)*C(3k,k)/(2k+1)}; a(n)=sum{k=0..n, C(3k/2,k/2)(1+(-1)^k)/(2(k+1))}. - Paul Barry, Nov 05 2006

G.f.: A(x) satisfies (x-1)*A(x)^3+(1-x)*A(x)-x = 0. - Mark van Hoeij, Apr 16 2013

Representation in terms of special values of hypergeometric function of type 4F3, in Maple notation a(n) = hypergeom([-n/2, (1-n)/2, 1/3, 2/3], [1, 1/2, 3/2], 27/4), n=0,1,2... . - Karol A. Penson, Jun 20 2013

Recurrence (for offset 0): -23*(n-3)*(n-2)*a(n-4) + 19*(n-2)*(2*n-3)*a(n-3) - 3*(n-2)*n*a(n-2) - 8*n*(2*n-1)*a(n-1) + 4*n*(n+1)*a(n) = 0. - Vaclav Kotesovec, Jun 28 2013

Asymptotics (for offset 0): a(n) ~ sqrt(3/Pi)*sqrt(1/2+1/(3*sqrt(3)))^3 * (1+3/2*sqrt(3))^n/n^(3/2). - Vaclav Kotesovec, Jun 28 2013

G.f.: 1/(1-x - 3*x^2/(S(0)*(1-x))), where S(k) = 4*k+3 - 3*x^2*(3*k+4)*(6*k+5)/( 2*(1-x)^2*(4*k+5) - 3*x^2*(3*k+5)*(6*k+7)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2013

Maple

series(RootOf((x-1)*A^3+(1-x)*A-x, A), x=0, 30); # Mark van Hoeij, Apr 16 2013

Mathematica

Table[Sum[Binomial[n, 2*k]*Binomial[3*k, k]/(2*k+1), {k, 0, Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 28 2013 *)

Prog

(PARI) Vec( serreverse(x*(1-x)*(1+x)/(1+x-x^3) +O(x^66) ) ) \\ Joerg Arndt, Sep 12 2013

Crossrefs

Sequence in context: A148097 A148098 A210043 * A101488 A279544 A245898

Adjacent sequences: A049127 A049128 A049129 * A049131 A049132 A049133

Keyword

nonn

Author

Olivier Gérard

Status

approved