A231691 Cardinalities of the symmetric operad of dotted red and white trees.

1, 6, 74, 1476, 41032, 1464672, 63865328, 3290120832, 195537380704, 13169097667584, 991181618539136, 82450282595311104, 7511417235983147008, 743790032122343820288, 79541198937597284060672, 9136079502141558495310848, 1121720442822518015112749056, 146607501639123412303738884096, 20322509742114322789584125210624, 2978025324234142178848508363882496

Offset: 1

Author

N. J. A. Sloane, Nov 14 2013

Keywords

• nonn

Examples

			A(x) = x + 6*x^2/2! + 74*x^3/3! + 1476*x^4/4! + 41032*x^5/5! + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..300
• F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Crossrefs

Cf. A052878, A231690.

Programs

Maple

S:= series(RootOf(y=-x-ln((1+x)/(1+3*x+x^2)),x),y,21):
seq(coeff(S,y,n)*n!,n=1..21); # Robert Israel, Sep 27 2018

Mathematica

terms = 20; (CoefficientList[InverseSeries[Log[x^2 + 3x + 1] - Log[1+x] - x + O[x]^(terms+1)], x]*Range[0, terms]!) // Rest (* Jean-François Alcover, Sep 16 2018, after Gheorghe Coserea *)

PARI

N=21; x = 'x + O('x^N); Vec(serlaplace(serreverse(log(x^2+3*x+1) - log(1+x) - x))) \\ Gheorghe Coserea, Jan 18 2017

Formula

E.g.f. A(x) satisfies -A(x) - g(-A(x)) = x where g is the E.g.f. of A052878. - Gheorghe Coserea, Jan 18 2017, edited by Robert Israel, Sep 27 2018

a(n) ~ sqrt((5 + 7*s + 3*s^2) / (7 + 13*s + 5*s^2)) * n^(n-1) / ((log((1+3*s+s^2)/(1+s))-s)^(n - 1/2) * exp(n)), where s = A060006 - 1 = -1 + (27/2 - 3*sqrt(69)/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3). - Vaclav Kotesovec, Apr 21 2020

Extensions

Offset changed and more terms from Gheorghe Coserea, Jan 15 2017