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
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
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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
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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 *) -
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
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 Jun Yan 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
G.f. A(x) = 1 + series_reversion( x/((1 + 3*x)*(1 + x)^2) ). - Peter Bala, Sep 10 2024
Extensions
Name clarified by Jun Yan, Apr 12 2024