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

0,3

Comments

See Novelli-Thibon (2014) for precise definition.

Links

Table of n, a(n) for n=0..21.

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.

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

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 *)

Crossrefs

Sequence in context: A025551 A101600 A233668 * A227169 A365184 A135148

Adjacent sequences: A243691 A243692 A243693 * A243695 A243696 A243697

Keyword

nonn,changed

Author

N. J. A. Sloane, Jun 14 2014

Extensions

More terms from Jun Yan, Apr 12 2024

Status

approved