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
Keywords
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.
Programs
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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
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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 *)
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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