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.
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.
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.
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
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 *)
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
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Oct 12 2003
STATUS
approved