A230008 - OEIS

%I #50 Jul 05 2020 07:58:42

%S 1,1,3,11,51,295,2055,16715,155355,1624255,18868575,241112675,

%T 3361168275,50760093175,825541973175,14385306834875,267379330468875,

%U 5280383597275375,110414649582951375,2437075699774353875,56622329782817431875,1381324610464997374375,35302637257323023454375

%N E.g.f. A(x) satisfies: A'(x) = -1 + A(x) + A(x)^2.

%C Counts binary free multilabeled increasing trees with m labels. - _Markus Kuba_, Nov 19 2014

%H Vaclav Kotesovec, <a href="/A230008/b230008.txt">Table of n, a(n) for n = 0..200</a>

%H Markus Kuba, Alois Panholzer, <a href="http://arxiv.org/abs/1411.4587">Combinatorial families of multilabelled increasing trees and hook-length formulas</a>, arXiv:1411.4587 [math.CO], (17-November-2014); see p. 29

%F E.g.f. A(x) satisfies:

%F (1) A(x) = 1 + Series_Reversion( Integral 1/(1 + 3*x + x^2) dx ).

%F (2) A(x) = exp(x + Integral A(x) - 1/A(x) dx).

%F (3) log(A(x)) = Series_Reversion( Integral 1/(1+2*sinh(x)) dx ).

%F (4) log(A(x)) = Series_Reversion( log( (sqrt(5)+2) * (Phi*exp(x) - 1)/(exp(x) + Phi) ) / sqrt(5) ) where Phi = (sqrt(5)+1)/2.

%F O.g.f.: 1 + x/(1-3*x - 1*2*x^2/(1-6*x - 2*3*x^2/(1-9*x - 3*4*x^2/(1-12*x - 4*5*x^2/(1-15*x - 5*6*x^2/(1- .../(1-3*n*x - n*(n+1)*x^2/(1- ...))))))) (continued fraction).

%F E.g.f.: A(x) = (2*(sqrt(5) - 2)*exp(sqrt(5)*x) + sqrt(5) - 1)/((3*sqrt(5) - 7)*exp(sqrt(5)*x) + 2). - _Vaclav Kotesovec_, Dec 20 2013

%F a(n) ~ n! * 5^((n+1)/2)/(arccosh(7/2))^(n+1). - _Vaclav Kotesovec_, Dec 20 2013

%F For n>=1: a(n) = 5^((n+1)/2) * sum(k>=1, k^n *((7-3*sqrt(5))/2)^k ). - _Markus Kuba_, Nov 19 2014

%F a(0) = a(1) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1). - _Ilya Gutkovskiy_, Jul 05 2020

%e G.f. = 1 + x + 3*x^2 + 11*x^3 + 51*x^4 + 295*x^5 + 2055*x^6 + 16715*x^7 + ...

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 11*x^3/3! + 51*x^4/4! + 295*x^5/5! +...

%e where A(x)^2 = 1 + 2*x + 8*x^2/2! + 40*x^3/3! + 244*x^4/4! + 1760*x^5/5! +...

%e also log(A(x)) = x + 2*x^2/2! + 4*x^3/3! + 10*x^4/4! + 44*x^5/5! + 290*x^6/6! +...

%e and 1/A(x) = 1 - x - x^2/2! + x^3/3! + 7*x^4/4! + 5*x^5/5! - 85*x^6/6! +...

%t FullSimplify[CoefficientList[Series[(2*(Sqrt[5] - 2)*E^(Sqrt[5]*x) + Sqrt[5] - 1)/((3*Sqrt[5] - 7)*E^(Sqrt[5]*x) + 2), {x, 0, 15}], x] * Range[0, 15]!] (* _Vaclav Kotesovec_, Dec 20 2013 *)

%t FullSimplify[Flatten[{1, Table[5^((n+1)/2)*PolyLog[-n,(7-3*Sqrt[5])/2], {n, 1, 20}]}]] (* _Vaclav Kotesovec_, Nov 19 2014 after _Markus Kuba_ *)

%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(x+intformal(A-1/A+x*O(x^n)))); n!*polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(A,X=x+x*O(x^n)); A=exp(serreverse(intformal(1/(1+2*sinh(X))))); n!*polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=1+serreverse(intformal(1/(1+3*x+x^2 +x*O(x^n))))); n!*polcoeff(A,n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A080635.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 19 2013