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A243694 Number of Hyposylvester classes of 4-multiparking functions of length n.

%I A243694 #29 Apr 13 2024 01:59:36
%S A243694 1,1,6,45,382,3498,33696,336549,3453750,36197694,385817700,4169274354,
%T A243694 45573898860,503014992340,5598239469972,62754598454805,
%U A243694 707899472049702,8029846915852662,91534356644739300,1048036064453687814,12047350849047152388,138984261578842304268
%N A243694 Number of Hyposylvester classes of 4-multiparking functions of length n.
%C A243694 See Novelli-Thibon (2014) for precise definition.
%H A243694 J.-C. Novelli and 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], 2014-2020. See Fig. 27.
%H A243694 Jun Yan, <a href="http://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.1.
%F A243694 a(n) = (1/n) * Sum_{k=0..n-1} 3^k * binomial(n,k) * binomial(3*n-k,2*n+1) for n > 0. - _Jun Yan_, Apr 12 2024
%F A243694 a(n) ~ sqrt(165 + 43*sqrt(33)) * (207 + 33*sqrt(33))^n / (sqrt(11*Pi) * n^(3/2) * 2^(5*n + 9/2)). - _Vaclav Kotesovec_, Apr 12 2024
%F A243694 a(n) = binomial(3*n, n) * hypergeom([1 - n, -n], [-3*n], -3) / (2*n + 1). - _Peter Luschny_, Apr 12 2024
%p A243694 a:= proc(n) option remember; `if`(n<2, 1, (3*(759*n^3-1725*n^2+1174*n-240)*
%p A243694       a(n-1)-54*(n-2)*(11*n-3)*(2*n-3)*a(n-2))/(8*(2*n+1)*(11*n-14)*n))
%p A243694     end:
%p A243694 seq(a(n), n=0..23);  # _Alois P. Heinz_, Apr 12 2024
%t A243694 a[n_] :=  Binomial[3*n, n] Hypergeometric2F1[1 - n, -n, -3 n, -3]  / (2 n + 1);
%t A243694 Table[a[n], {n, 0, 21}]  (* _Peter Luschny_, Apr 12 2024 *)
%K A243694 nonn
%O A243694 0,3
%A A243694 _N. J. A. Sloane_, Jun 14 2014
%E A243694 More terms from _Jun Yan_, Apr 12 2024