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A088716 G.f. satisfies: A(x) = 1 + x*A(x)*d/dx[x*A(x)] = 1 + x*A(x)^2 + x^2*A(x)*A'(x). 33
1, 1, 3, 14, 85, 621, 5236, 49680, 521721, 5994155, 74701055, 1003125282, 14437634276, 221727608284, 3619710743580, 62605324014816, 1143782167355649, 22014467470369143, 445296254367273457, 9444925598142843970 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

M. J. H. Al-Kaabi, Monomial Bases for free pre-Lie algebras, Sem. Lothar. Comb. 71 (2014) B71b

FORMULA

a(n) = Sum_{k=1..n} k*a(k-1)*a(n-k) for n>=1 with a(0)=1.

Forms column 0 of triangle T=A112911, where the matrix inverse satisfies: [T^-1](n,k) = -(k+1)*T(n-1,0) for n>k>=0.

Self-convolution is A112916, where a(n) = (n+1)/2*A112916(n-1) for n>0.

G.f.: A(x) = serreverse(x/f(x))/x where f(x) is the g.f. of A088715.

O.g.f.: A(x) = log(G(x))/x where G(x) is the e.g.f. of A182962 given by:

. G(x) = exp( x/(1 - x*G'(x)/G(x)) ). [From Paul D. Hanna, Jan 01 2011]

O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) / A(x) = 0 for n>0. - Paul D. Hanna, May 25 2018

O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (1 - n*x) = 0 for n>0. - Paul D. Hanna, Jul 24 2019

From Paul D. Hanna, Jul 20 2018 (Start):

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

* [x^n] exp(-n * x*A(x)) * (2 - 1/A(x)) = 0 for n >= 1.

* [x^n] exp(-n^2 * x*A(x)) * (n + 1 - n/A(x)) = 0 for n >= 1.

* [x^n] exp(-n^(p+1) * x*A(x)) * (n^p + 1 - n^p/A(x)) = 0 for n>=1 and for fixed integer p >= 0. (End)

a(n) ~ c * n! * n^2, where c = 0.21795078944715106549282282244231982088... (see A238223). - Vaclav Kotesovec, Feb 21 2014

MAPLE

a:= proc(n) option remember; `if`(n=0, 1, add(

      a(j)*a(n-j-1)*(j+1), j=0..n-1))

    end:

seq(a(n), n=0..25);  # Alois P. Heinz, Aug 10 2017

MATHEMATICA

a=ConstantArray[0, 21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = Sum[k*a[[n-k+1]]*a[[k]], {k, 1, n}], {n, 2, 20}]; a (* Vaclav Kotesovec, Feb 21 2014 *)

m = 20; A[_] = 0;

Do[A[x_] = 1 + x A[x]^2 + x^2 A[x] A'[x] + O[x]^m // Normal, {m}];

CoefficientList[A[x], x] (* Jean-Fran├žois Alcover, Feb 18 2020 *)

PROG

(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, (k+1)*a(k)*a(n-k-1)))

(PARI) {a(n)=local(G=1+x); for(i=1, n, G=exp(x/(1 - x*deriv(G)/G+x*O(x^n)))); polcoeff(log(G)/x, n)} \\ Paul D. Hanna, Jan 01 2011

CROSSREFS

Cf. A112916 (A^2), A112911, A112912, A112913, A112914.

Cf. A088715, A182962, A112915, A218222, A238223.

Cf. A300736, A300987, A300989, A300991, A300993.

Sequence in context: A160881 A263187 A213628 * A005189 A331608 A331615

Adjacent sequences:  A088713 A088714 A088715 * A088717 A088718 A088719

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 12 2003

STATUS

approved

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Last modified April 14 19:59 EDT 2021. Contains 342954 sequences. (Running on oeis4.)