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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).

%I A088716 #59 Nov 03 2024 12:25:22
%S A088716 1,1,3,14,85,621,5236,49680,521721,5994155,74701055,1003125282,
%T A088716 14437634276,221727608284,3619710743580,62605324014816,
%U A088716 1143782167355649,22014467470369143,445296254367273457,9444925598142843970
%N 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).
%H A088716 Vincenzo Librandi, <a href="/A088716/b088716.txt">Table of n, a(n) for n = 0..200</a>
%H A088716 M. J. H. Al-Kaabi, <a href="https://www.emis.de/journals/SLC/wpapers/s71mahdi.html">Monomial Bases for free pre-Lie algebras</a>, Sem. Lothar. Comb. 71 (2014) B71b.
%H A088716 Michael Borinsky, Gerald V. Dunne, and Karen Yeats, <a href="https://www.arxiv.org/abs/2408.15883">Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations</a>, arXiv:2408.15883 [math-ph], 2024. See p. 13.
%H A088716 Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 11.
%F A088716 a(n) = Sum_{k=1..n} k*a(k-1)*a(n-k) for n>=1 with a(0)=1.
%F A088716 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.
%F A088716 Self-convolution is A112916, where a(n) = (n+1)/2*A112916(n-1) for n>0.
%F A088716 G.f.: A(x) = serreverse(x/f(x))/x where f(x) is the g.f. of A088715.
%F A088716 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]
%F A088716 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
%F A088716 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
%F A088716 From _Paul D. Hanna_, Jul 20 2018 (Start):
%F A088716 O.g.f. A(x) satisfies:
%F A088716 * [x^n] exp(-n * x*A(x)) * (2 - 1/A(x)) = 0 for n >= 1.
%F A088716 * [x^n] exp(-n^2 * x*A(x)) * (n + 1 - n/A(x)) = 0 for n >= 1.
%F A088716 * [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)
%F A088716 a(n) ~ c * n! * n^2, where c = 0.21795078944715106549282282244231982088... (see A238223). - _Vaclav Kotesovec_, Feb 21 2014
%p A088716 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A088716       a(j)*a(n-j-1)*(j+1), j=0..n-1))
%p A088716     end:
%p A088716 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 10 2017
%t A088716 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 *)
%t A088716 m = 20; A[_] = 0;
%t A088716 Do[A[x_] = 1 + x A[x]^2 + x^2 A[x] A'[x] + O[x]^m // Normal, {m}];
%t A088716 CoefficientList[A[x], x] (* _Jean-François Alcover_, Feb 18 2020 *)
%t A088716 a[1]:=1; a[2]:=1; a[n_]:=a[n]=n/2 Sum[a[k] a[n-k], {k,1,n-1}];
%t A088716 Map[a,Range[20]] (* _Oliver Seipel_, Nov 03 2024 ,after Schröder 1870 *)
%o A088716 (PARI) a(n)=if(n==0,1,sum(k=0,n-1,(k+1)*a(k)*a(n-k-1)))
%o A088716 (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
%Y A088716 Cf. A112916 (A^2), A112911, A112912, A112913, A112914.
%Y A088716 Cf. A088715, A182962, A112915, A218222, A238223.
%Y A088716 Cf. A300736, A300987, A300989, A300991, A300993.
%K A088716 nonn
%O A088716 0,3
%A A088716 _Paul D. Hanna_, Oct 12 2003