A243675 Number of hypoplactic classes of 3-parking functions of length n.

1, 1, 7, 67, 741, 8909, 113107, 1492103, 20251945, 280978681, 3967031839, 56811348235, 823250855181, 12049087175493, 177857857845675, 2644773866954255, 39581787842355409, 595745692419162737, 9011736489133233463, 136932249972928786387, 2089082351509217490613

Offset

0,3

Comments

See Novelli-Thibon (2014) for precise definition.

This is almost certainly the sequence of small 4-Schroeder numbers as defined by Yang-Jiang (2021). It would be nice to have a proof. Then we could confirm Weiner's conjectured formula, and extend the sequence. Yang & Jiang (2021) give an explicit formula for the small m-Schroeder numbers in Theorems 2.4 and 2.9. - N. J. A. Sloane, Mar 28 2021

This is also the small 4-Schroeder numbers defined by Yang and Jiang (2021) in Theorems 2.4 and 2.9. - Jun Yan, Apr 13 2024

References

Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

Links

Table of n, a(n) for n=0..20.

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. 23.

Jun Yan, Results on pattern avoidance in parking functions, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.

Formula

a(n) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(3i+j, n)*binomial(n+1,i)/(n+1) (conjectured). - Michael D. Weiner, May 25 2017

From Jun Yan, Apr 13 2024 : (Start)

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

Let D(n) be the set of 3-Dyck paths with n up-steps of size 3, 3n down-steps of size 1 and never go below the x-axis. For every d in D(n), let peak(d) be the number of peaks in d. Then a(n) = Sum_{d in D(n)}2^{peak(d) - 1}. (End)

a(n) = hypergeom([1 - n, -3*n], [2], 2). - Peter Luschny, Apr 13 2024

D-finite with recurrence -15*n*(3*n-1)*(3*n+1)*a(n) +(43*n^3+5403*n^2-8482*n+3228)*a(n-1) +6*(6039*n^3-33372*n^2+60401*n-35858) *a(n-2) +9*(-689*n^3+5938*n^2-17157*n+16616)*a(n-3) +27*(3*n-10)*(3*n-11)*(n-4)*a(n-4)=0. - R. J. Mathar, Apr 14 2024

D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*(35*n^2-98*n+68)*a(n) +(-15610*n^5+67123*n^4-106824*n^3+77633*n^2-25514*n+3000)*a(n-1) +3*(n-2)*(3*n-4)*(3*n-5)*(35*n^2-28*n+5)*a(n-2)=0. - R. J. Mathar, Apr 14 2024

Maple

a := proc(n) option remember; if n <= 1 then return 1 fi;
-((945*n^5 - 5481*n^4 + 11685*n^3 - 11091*n^2 + 4470*n - 600)*a(n - 2) +
(-15610*n^5 + 67123*n^4 - 106824*n^3 + 77633*n^2 - 25514*n + 3000)*a(n - 1)) /
(945*n^5 - 2646*n^4 + 1731*n^3 + 294*n^2 - 204*n) end:
seq(a(n), n = 0..20);  # Peter Luschny, Apr 13 2024

Mathematica

a[n_] := Hypergeometric2F1[1 - n, -3 n, 2, 2];
Table[a[n], {n, 0, 20}]  (* Peter Luschny, Apr 13 2024 *)

Crossrefs

The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

Apparently, a(n) = A144097/2, apart from the initial term. - N. J. A. Sloane, Mar 28 2021 [This is for n > 0 indeed the case. - Jun Yan, Apr 13 2024]

Sequence in context: A199756 A038386 A371398 * A364924 A082578 A291814

Adjacent sequences: A243672 A243673 A243674 * A243676 A243677 A243678

Keyword

nonn,changed

Author

N. J. A. Sloane, Jun 14 2014

Extensions

Added a(0) = 1. - N. J. A. Sloane, Mar 28 2021

More terms from Jun Yan, Apr 13 2024

Status

approved