%I #25 Feb 24 2016 11:14:39
%S 1,1,4,27,254,3048,44328,755681,14750646,323999500,7901623624,
%T 211690439030,6176393964684,194847458672328,6606138879434128,
%U 239466033046020357,9239284257332493478,377948418993992417644,16335430070738649950536,743711790322786003051882
%N Number of meta-Sylvester classes of 2-multiparking functions of length n.
%C See Novelli-Thibon (2014) for precise definition.
%C Sum over all Dyck paths of semilength n of products over all peaks p of (1+x_p-y_p), where x_p and y_p are the coordinates of peak p. - _Alois P. Heinz_, May 27 2015
%H Alois P. Heinz, <a href="/A243696/b243696.txt">Table of n, a(n) for n = 0..400</a>
%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 28.
%F G.f.: 1/(1-x) = Sum_{n>=0} a(n) * x^n*(1-x)^n / Product_{k=1..n} (1 + 2*k*x). - _Paul D. Hanna_, Jun 14 2014
%p b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (1+x-y), 1) +
%p b(x-1, y+1, true)))
%p end:
%p a:= n-> b(2*n, 0, false):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 27 2015
%t b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x==0, 1, b[x-1, y-1, False]*If[t, 1+x-y, 1] + b[x-1, y+1, True]]]; a[n_] := b[2*n, 0, False]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Feb 24 2016, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=if(n<0, 0, polcoeff(1/(1-x+x*O(x^n)) - sum(k=1, n-1, a(k)*x^k*(1-x)^k/prod(j=0, k, 1+2*j*x+x*O(x^n))), n))}
%o for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jun 14 2014
%Y Cf. A132624, A243697, A243698, A243699.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jun 14 2014
%E Offset changed to 0 by _Paul D. Hanna_, Jun 14 2014