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%I #25 Nov 08 2017 07:44:35
%S 1,0,5,50,634,9475,163843,3226213,71430404,1759835599,47821543220,
%T 1422411027534,46002758077823,1608256429511163,60463005173005523,
%U 2433267830904336072,104394054462487756061,4757234883237958801214,229506935072122869176226
%N Conjecturally, number of generators of degree n of the Hopf algebra of parking functions, regarded as a dendriform trialgebra.
%H Alois P. Heinz, <a href="/A141316/b141316.txt">Table of n, a(n) for n = 1..150</a>
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://dx.doi.org/10.4064/fm193-3-1">Hopf algebras and dendriform structures arising from parking functions</a>, Fundamenta Math. 193 (2007), 189-241.
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions</a>, arXiv:0806.3682 [math.CO], 2008; Discrete Math. 310 (2010), no. 24, 3584-3606. See Eq. (120).
%F G.f.: (f(t)-1)/(2f(t)^2-f(t)) where f(t) = 1 + Sum_{n>=1} (n+1)^(n-1)*t^n.
%F a(n) ~ exp(1) * n^(n-1). - _Vaclav Kotesovec_, Sep 10 2014
%p f:= proc(N) 1+add((n+1)^(n-1)*t^n, n=1..N) end: g:= proc(N) taylor( (f(N)-1)/ (2*f(N)^2-f(N)), t, N+1) end: a:= proc(n) coeff(g(n), t, n) end: seq(a(n), n=1..20);
%t terms = 20; f[t_] = 1 + Sum[(n + 1)^(n - 1)*t^n, {n, 1, terms}]; (1/t)* (f[t] - 1)/(2*f[t]^2 - f[t]) + O[t]^terms // CoefficientList[#, t]& (* _Jean-François Alcover_, Nov 08 2017, after _Vaclav Kotesovec_ *)
%o (PARI) lista(m) = {t = u + O(u^(m+1)); P = 1+sum(n=1, m, (n+1)^(n-1)*t^n); Q = (P-1)/(2*P^2-P); for (n=1, m, print1(polcoeff(Q, n, u), ", "));} \\ _Michel Marcus_, Feb 12 2013
%Y Cf. A122705, A122708, A185281.
%K nonn
%O 1,3
%A Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
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