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

1, 1, 6, 45, 382, 3498, 33696, 336549, 3453750, 36197694, 385817700, 4169274354, 45573898860, 503014992340, 5598239469972, 62754598454805, 707899472049702, 8029846915852662, 91534356644739300, 1048036064453687814, 12047350849047152388, 138984261578842304268

Offset: 0

Author

N. J. A. Sloane, Jun 14 2014

Keywords

• nonn

Comments

See Novelli-Thibon (2014) for precise definition.

Links

• J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020. See Fig. 27.
• Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.1.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 1, (3*(759*n^3-1725*n^2+1174*n-240)*
      a(n-1)-54*(n-2)*(11*n-3)*(2*n-3)*a(n-2))/(8*(2*n+1)*(11*n-14)*n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2024

Mathematica

a[n_] :=  Binomial[3*n, n] Hypergeometric2F1[1 - n, -n, -3 n, -3]  / (2 n + 1);
Table[a[n], {n, 0, 21}]  (* Peter Luschny, Apr 12 2024 *)

Formula

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

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

a(n) = binomial(3*n, n) * hypergeom([1 - n, -n], [-3*n], -3) / (2*n + 1). - Peter Luschny, Apr 12 2024

Extensions

More terms from Jun Yan, Apr 12 2024