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).

1, 1, 3, 14, 85, 621, 5236, 49680, 521721, 5994155, 74701055, 1003125282, 14437634276, 221727608284, 3619710743580, 62605324014816, 1143782167355649, 22014467470369143, 445296254367273457, 9444925598142843970

Offset: 0

Author

Paul D. Hanna, Oct 12 2003

Keywords

• nonn

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.
• Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 13.
• Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 11.

Crossrefs

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

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

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

Programs

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-François Alcover, Feb 18 2020 *)
a[1]:=1; a[2]:=1; a[n_]:=a[n]=n/2 Sum[a[k] a[n-k], {k,1,n-1}];
Map[a,Range[20]] (* Oliver Seipel, Nov 03 2024 ,after Schröder 1870 *)

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

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)) ). [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