A243693 Number of Hyposylvester classes of 3-multiparking functions of length n.

1, 1, 5, 32, 233, 1833, 15180, 130392, 1151057, 10378883, 95182445, 885053524, 8324942620, 79071217228, 757310811912, 7305728683824, 70923966744609, 692370887676567, 6792525607165935, 66933512163735000, 662190712902022017, 6574831459429388169, 65494637699437417584

Offset

0,3

Links

Table of n, a(n) for n=0..22.

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

From Seiichi Manyama, Aug 12 2023: (Start)

The following statements are equivalent:

The g.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - 2*x*A(x)^2).

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n, k) * binomial(2*n+k+1, n) / (2*n + k + 1).

a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n, k) * binomial(2*n, k-1) for n > 0.

a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n, k)*binomial(3*n-k, n-1-k) for n > 0.

(End)

The above formula is proved in Theorem 4.1 of the second link to be the number of Hyposylvester classes of 3-multiparking functions of length n. - Jun Yan, Apr 12 2024

a(n) ~ 2^(5*n+1) / (sqrt(5*Pi) * n^(3/2) * 3^(n+1)). - Vaclav Kotesovec, Apr 12 2024

a(n) = 3^(n - 1) * hypergeom([1 - n, -2*n], [2], 1/3) for n > 0. - Peter Luschny, Apr 12 2024

Maple

a := proc(n) option remember; if n <= 1 then return 1 fi;
(a(n - 2)*(-800*n^3 + 3024*n^2 - 3184*n + 672) + a(n - 1)*(3275*n^3 - 7467*n^2 +
5038*n - 1008))/(300*n^3 - 234*n^2 - 192*n) end:
seq(a(n), n = 0..22);  # Peter Luschny, Apr 13 2024

Mathematica

a[n_] := 3^(n - Boole[n>0]) Hypergeometric2F1[1 - n, -2 n, 2, 1/3];
Table[a[n], {n, 0, 22}]  (* Peter Luschny, Apr 12 2024 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1));  Seiichi Manyama, Aug 12 2023

Crossrefs

Cf. A243694, A243695.

Cf. A007564, A003168, A364923, A364924.

Sequence in context: A065071 A153396 A146965 * A364747 A053157 A102231

Adjacent sequences: A243690 A243691 A243692 * A243694 A243695 A243696

Keyword

nonn,changed

Author

N. J. A. Sloane, Jun 14 2014

Extensions

Name clarified by Jun Yan, Apr 12 2024

Status

approved