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A141316
Conjecturally, number of generators of degree n of the Hopf algebra of parking functions, regarded as a dendriform trialgebra.
3
1, 0, 5, 50, 634, 9475, 163843, 3226213, 71430404, 1759835599, 47821543220, 1422411027534, 46002758077823, 1608256429511163, 60463005173005523, 2433267830904336072, 104394054462487756061, 4757234883237958801214, 229506935072122869176226
OFFSET
1,3
LINKS
J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions, Fundamenta Math. 193 (2007), 189-241.
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008; Discrete Math. 310 (2010), no. 24, 3584-3606. See Eq. (120).
FORMULA
G.f.: (f(t)-1)/(2f(t)^2-f(t)) where f(t) = 1 + Sum_{n>=1} (n+1)^(n-1)*t^n.
a(n) ~ exp(1) * n^(n-1). - Vaclav Kotesovec, Sep 10 2014
MAPLE
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);
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
STATUS
approved