A230008 E.g.f. A(x) satisfies: A'(x) = -1 + A(x) + A(x)^2.

1, 1, 3, 11, 51, 295, 2055, 16715, 155355, 1624255, 18868575, 241112675, 3361168275, 50760093175, 825541973175, 14385306834875, 267379330468875, 5280383597275375, 110414649582951375, 2437075699774353875, 56622329782817431875, 1381324610464997374375, 35302637257323023454375

Offset

0,3

Comments

Counts binary free multilabeled increasing trees with m labels. - Markus Kuba, Nov 19 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], (17-November-2014); see p. 29

Formula

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

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

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

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

(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.

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).

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

a(n) ~ n! * 5^((n+1)/2)/(arccosh(7/2))^(n+1). - Vaclav Kotesovec, Dec 20 2013

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

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

Example

G.f. = 1 + x + 3*x^2 + 11*x^3 + 51*x^4 + 295*x^5 + 2055*x^6 + 16715*x^7 + ...
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 11*x^3/3! + 51*x^4/4! + 295*x^5/5! +...
where A(x)^2 = 1 + 2*x + 8*x^2/2! + 40*x^3/3! + 244*x^4/4! + 1760*x^5/5! +...
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! +...
and 1/A(x) = 1 - x - x^2/2! + x^3/3! + 7*x^4/4! + 5*x^5/5! - 85*x^6/6! +...

Mathematica

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 *)
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 *)

Prog

(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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A, X=x+x*O(x^n)); A=exp(serreverse(intformal(1/(1+2*sinh(X))))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+serreverse(intformal(1/(1+3*x+x^2 +x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A080635.

Sequence in context: A184819 A113712 A056199 * A007047 A182176 A244754

Adjacent sequences: A230005 A230006 A230007 * A230009 A230010 A230011

Keyword

nonn

Author

Paul D. Hanna, Dec 19 2013

Status

approved