A243676 Number of hypoplactic classes of 4-parking functions of length n.

1, 1, 9, 113, 1649, 26225, 440985, 7711009, 138792929, 2554489505, 47854963881, 909495557393, 17492724268369, 339846019830673, 6659441891042105, 131467175048437569, 2612224160086781889, 52201209713045788737, 1048450942860766632777, 21153308764742204273329, 428520989167282737342513

Offset

0,3

Comments

This is almost certainly the sequence of small 5-Schroeder numbers as defined by Yang-Jiang (2021). It would be nice to have a proof. Then we could confirm Weiner's conjectured formulas, 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 indeed the small 5-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+1) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(4*i+j, n)*binomial(n+1,i)/(n+1) (conjectured). - Michael D. Weiner, May 25 2017

a(n) = Sum_{i=1..n} binomial(4*n, i-1)*binomial(n, i)*2^(i-1)/n (conjectured). - Michael D. Weiner, Jul 24 2019 [This is correct for n>0 - Jun Yan, Apr 13 2024]

Let D(n) be the set of 4-Dyck paths with n up-steps of size 4, 4n 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). - Jun Yan, Apr 13 2024

a(n) = hypergeom([1 - n, -4*n], [2], 2). - Peter Luschny, Apr 13 2024

Maple

a := proc(n) option remember; if n <= 1 then return 1 fi;
-(a(n-2)*(-5440*n^7 + 42080*n^6 - 131548*n^5 + 212750*n^4 - 189160*n^3 + 90725*n^2 - 21387*n + 1890)+ a(n-1)*(-118660*n^7 + 739880*n^6 - 1876702*n^5 + 2492120*n^4 - 1855960*n^3 + 768230*n^2 - 161913*n + 13230)) / (5440*n^7 - 25760*n^6 + 43468*n^5 - 29510*n^4 + 4750*n^3 + 1945*n^2 - 468*n) end:
seq(a(n), n = 0..20);  # Peter Luschny, Apr 13 2024

Mathematica

a[n_] := Hypergeometric2F1[1 - n, -4 n, 2, 2];
Table[a[n], {n, 0, 20}]  (* Peter Luschny, Apr 13 2024 *)

Crossrefs

Appears to equal A260332(n)/2 for n > 0. - N. J. A. Sloane, Mar 28 2021

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

Sequence in context: A155624 A165224 A342296 * A012116 A362576 A356240

Adjacent sequences: A243673 A243674 A243675 * A243677 A243678 A243679

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