0,3

G.f. (offset 1) is series reversion of x-x^2-x^3.

Antidiagonal sums of triangle A104978 which has g.f. F(x,y) that satisfies: F = 1 + x*F^2 + x*y*F^3. - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005

a(n+1) is number of (2,3)-rooted trees on n nodes.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.

T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 211 (3.2.73-74)

T. D. Noe, Table of n, a(n) for n=0..100

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 395

Index entries for reversions of series

a(n)=sum(binomial(n+k, k)*binomial(k, n-k), k=ceil(n/2)..n)/(n+1); 5n(n+1) * a(n) = 11n(2n-1) * a(n-1) + 3(3n-2)(3n-4) * a(n-2) - Len Smiley (smiley(AT)math.uaa.alaska.edu)

G.f.: (4sin(asin((27x+11)/16)/3)-1)/(3x); - Paul Barry (pbarry(AT)wit.ie), Feb 02 2005

a(n) = Sum_{k=0..[n/2]} C(2*n-k, n+k)*C(n+k, k)/(n+1). - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005

a(3)=10 because a convex pentagon can be dissected in 5 ways into triangles (draw 2 diagonals from any of the 5 vertices) and in 5 ways into a triangle and a quadrilateral (draw any of the 5 diagonals).

Rest[CoefficientList[InverseSeries[Series[y - y^2 - y^3, {y, 0, 30}], x], x]]

(PARI) a(n)=if(n<0, 0, polcoeff(serreverse(x-x^2-x^3+x^2*O(x^n)), n+1))

(PARI) a(n)=if(n<0, 0, sum(k=0, n\2, (2*n-k)!/k!/(n-2*k)!)/(n+1)!)

(PARI) a(n)=sum(k=0, n\2, binomial(2*n-k, n+k)*binomial(n+k, k))/(n+1) (Hanna)

n*a(n)=A038112(n-1), n>0.

Cf. A104978.

Sequence in context: A151060 A151061 A109085 this_sequence A151062 A000902 A151063

Adjacent sequences: A000999 A001000 A001001 this_sequence A001003 A001004 A001005

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000

Revised by Emeric Deutsch and Len Smiley, Jun 05, 2005