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

%I A231691 #43 Apr 21 2020 07:38:32
%S A231691 1,6,74,1476,41032,1464672,63865328,3290120832,195537380704,
%T A231691 13169097667584,991181618539136,82450282595311104,7511417235983147008,
%U A231691 743790032122343820288,79541198937597284060672,9136079502141558495310848,1121720442822518015112749056,146607501639123412303738884096,20322509742114322789584125210624,2978025324234142178848508363882496
%N A231691 Cardinalities of the symmetric operad of dotted red and white trees.
%H A231691 Gheorghe Coserea, <a href="/A231691/b231691.txt">Table of n, a(n) for n = 1..300</a>
%H A231691 F. Chapoton, F. Hivert, J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
%F A231691 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
%F A231691 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
%e A231691 A(x) = x + 6*x^2/2! + 74*x^3/3! + 1476*x^4/4! + 41032*x^5/5! + ...
%p A231691 S:= series(RootOf(y=-x-ln((1+x)/(1+3*x+x^2)),x),y,21):
%p A231691 seq(coeff(S,y,n)*n!,n=1..21); # _Robert Israel_, Sep 27 2018
%t A231691 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_ *)
%o A231691 (PARI)
%o A231691 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
%Y A231691 Cf. A052878, A231690.
%K A231691 nonn
%O A231691 1,2
%A A231691 _N. J. A. Sloane_, Nov 14 2013
%E A231691 Offset changed and more terms from _Gheorghe Coserea_, Jan 15 2017