A141200 G.f. satisfies: A(x) = x + A(A(x)^2).
1, 1, 2, 6, 20, 72, 272, 1065, 4282, 17576, 73344, 310226, 1327036, 5730948, 24952776, 109417672, 482779032, 2141832444, 9548501992, 42753897498, 192184437012, 866963862560, 3923596330784, 17809292215406, 81055344516420
Offset: 1
Keywords
Examples
G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 72*x^6 + 272*x^7 +... The g.f. satisfies the series: A(x) = x + A(x^2) + d/dx A(x^2)^2/2! + d^2/dx^2 A(x^2)^3/3! + d^3/dx^3 A(x^2)^4/4! +... as well as the logarithmic series: log(A(x)/x) = A(x^2)/x + [d/dx A(x^2)^3/x]/2! + [d^2/dx^2 A(x^2)^3/x]/3! + [d^3/dx^3 A(x^2)^4/x]/4! +... Related expansions. A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 208*x^7 + 804*x^8 +... The series reversion of A(x) equals x - A(x^2), therefore A(x - x^2 - x^4 - 2*x^6 - 6*x^8 - 20*x^10 - 72*x^12 -...) = x. Let G(x) = A(x)^2 then G(G(x)) = x^4 + 4*x^5 + 16*x^6 + 64*x^7 + 260*x^8 + 1072*x^9 +... G(G(G(x))) = x^8 + 8*x^9 + 48*x^10 + 256*x^11 + 1290*x^12 +... where A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..100
- Li Guo, Yunnan Li, Braided dendriform and tridendriform algebras and braided Hopf algebras of planar trees, arXiv:1906.06454 [math.QA], 2019.
- Yunnan Li, Li Guo, Braided dendriform and tridendriform algebras, and braided Hopf algebras of rooted trees, Journal of Algebraic Combinatorics (2020).
Crossrefs
Cf. A141201.
Cf. A178852. [From Paul D. Hanna, Jun 18 2010]
Programs
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Mathematica
terms = 25; A[] = 0; Do[A[x] = x + A[A[x]^2] + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *) a[ n_] := If[ n < 0, 0, Module[{A = 0}, Do[ A = Normal[A] + x O[x]^k; A = x + (A /. x -> A^2), {k, n}]; SeriesCoefficient[ A, {x, 0, n}]]]; (* Michael Somos, Jul 16 2018 *)
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Maxima
Co(n, k, F):=if k=1 then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F), i, 0, n-k); a(n):=if n=1 then 1 else sum(if 2*k>n then 0 else Co(n, 2*k, a)*a(k), k, 1, n); makelist(a(n), n, 1, 10); /*Vladimir Kruchinin, Aug 02 2011 */
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Maxima
T(n,m):=if n=m then 1 else kron_delta(n,m)+sum(binomial(m,j)*sum(if 2*k<=n-j then T(n-j,2*k)*T(k,m-j) else 0,k,m-j,n-j),j,0,m-1); makelist(T(n,1),n,1,12); /* Vladimir Kruchinin, May 02 2012 */
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PARI
{a(n)=local(A=x+x^2);for(i=0,n,A=serreverse(x-subst(A,x,x^2+x^2*O(x^n)))) ;polcoeff(A,n)} -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, subst(A,x,x^2+x*O(x^n))^m)/m!)+x*O(x^n)); polcoeff(A, n)} for(n=1,25,print1(a(n),", ")) -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, subst(A,x,x^2+x*O(x^n))^m/x)/m!)+x*O(x^n))); polcoeff(A, n)} for(n=1,25,print1(a(n),", ")) -
PARI
{a(n) = my(A); if( n<0, 0, for(k=1, n, A = truncate(A) + x * O(x^k); A = x + subst(A, x, A^2)); polcoeff(A, n))}; /* Michael Somos, Jul 16 2018 */
Formula
G.f. A(x) satisfies:
(1) A(x - A(x^2)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x^2)^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x^2)^n/x / n! ).
(4) A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +... where G(x) = A(x)^2 = g.f. of A141201.
Contribution from Paul D. Hanna, Jun 18 2010: (Start)
Derivative of g.f. A(x) satisfies A'(x) = 1/(1 - 2*A(x)*A'(A(x)^2)).
Radius of convergence, r, and related values:
. r = 0.206450159053688924498041214308637032933597292895284203439137...
. A(r) = 0.350063281326319514237505104302392755865233862157808469329...
where r = A(r) - A(A(r)^2);
. A(A(r)^2) = 0.1436131222726305897394638899937557229316365692625242...
. A'(A(r)^2) = 1.428313184135166508863259733728425402891099463888244...
where A'(A(r)^2) = 1/(2*A(r)).
G.f. of A178852 is V(x) = x/(x - A(x^2)) where:
V'(A(r)) = 1/r,
V(A(x)) = A(x)/x and A(x/V(x)) = x.
(End)
Let B(x) = Sum_{n>=1} a(n)*x^(2*n), then B(x) = x^2 + B(B(x)). [From Paul D. Hanna, Jul 15 2011]
a(n) ~ c / (r^n * n^(3/2)), where c = 0.073344948246606003114646... . - Vaclav Kotesovec, Dec 02 2014
Comments