cp's OEIS Frontend

A243695 Number of Hyposylvester classes of 5-multiparking functions of length n.

%I A243695 #39 Sep 11 2024 05:48:06
%S A243695 1,1,7,60,579,6017,65732,744264,8656795,102819507,1241838271,
%T A243695 15205587136,188320591092,2354971302700,29693879866840,
%U A243695 377104836064720,4819271465838795,61930407776801015,799765007716515125,10373651783800459340,135089139660222638795
%N A243695 Number of Hyposylvester classes of 5-multiparking functions of length n.
%C A243695 See Novelli-Thibon (2014) for precise definition.
%H A243695 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 A243695 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 A243695 a(n) = (1/n) * Sum_{k=0..n-1} 4^k * binomial(n,k) * binomial(3*n-k,2*n+1) for n > 0. - _Jun Yan_, Apr 12 2024
%F A243695 a(n) = Sum_{k=0..n} 5^k * (-4)^(n-k) * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1). - _Alois P. Heinz_, Apr 12 2024
%F A243695 a(n) = (-4)^n * CatalanNumber(n) * hypergeom([-n, 2*n + 1], [n + 2], 5/4). - _Peter Luschny_, Apr 12 2024
%F A243695 a(n) ~ sqrt(779 + 201*sqrt(41)) * (299 + 41^(3/2))^n / (sqrt(41*Pi) * n^(3/2) * 2^(3*n + 5/2) * 5^(n+1)). - _Vaclav Kotesovec_, Apr 12 2024
%F A243695 From _Peter Bala_, Sep 08 2024: (Start)
%F A243695 G.f. A(x) = 1 + series_reversion( x/((1 + 5*x)*(1 + x)^2) ).
%F A243695 A(x) = 1 + x*(5*A(x)^3 - 4*A(x)^2). (End)
%p A243695 a := proc(n) option remember; if n <= 1 then return 1 fi;
%p A243695 (a(n - 2)*(-10496*n^3 + 39552*n^2 - 41344*n + 8448) + a(n - 1)*(12259*n^3 -
%p A243695 27807*n^2 + 19058*n - 3960)) / (820*n^3 - 630*n^2 - 520*n) end:
%p A243695 seq(a(n), n = 0..20);  # _Peter Luschny_, Apr 13 2024
%t A243695 a[n_] := (-4)^n * CatalanNumber[n] Hypergeometric2F1[-n, 2 n + 1, n + 2, 5/4];
%t A243695 Table[a[n], {n, 0, 20}]  (* _Peter Luschny_, Apr 12 2024 *)
%Y A243695 Cf. A243693, A243694.
%K A243695 nonn,easy
%O A243695 0,3
%A A243695 _N. J. A. Sloane_, Jun 14 2014
%E A243695 More terms from _Jun Yan_, Apr 12 2024