A243695 Number of Hyposylvester classes of 5-multiparking functions of length n.
1, 1, 7, 60, 579, 6017, 65732, 744264, 8656795, 102819507, 1241838271, 15205587136, 188320591092, 2354971302700, 29693879866840, 377104836064720, 4819271465838795, 61930407776801015, 799765007716515125, 10373651783800459340, 135089139660222638795
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)*(-10496*n^3 + 39552*n^2 - 41344*n + 8448) + a(n - 1)*(12259*n^3 - 27807*n^2 + 19058*n - 3960)) / (820*n^3 - 630*n^2 - 520*n) end: seq(a(n), n = 0..20); # Peter Luschny, Apr 13 2024
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Mathematica
a[n_] := (-4)^n * CatalanNumber[n] Hypergeometric2F1[-n, 2 n + 1, n + 2, 5/4]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Apr 12 2024 *)
Formula
a(n) = (1/n) * Sum_{k=0..n-1} 4^k * binomial(n,k) * binomial(3*n-k,2*n+1) for n > 0. - Jun Yan, Apr 12 2024
a(n) = Sum_{k=0..n} 5^k * (-4)^(n-k) * binomial(n,k) * binomial(2*n+k+1,n) / (2*n+k+1). - Alois P. Heinz, Apr 12 2024
a(n) = (-4)^n * CatalanNumber(n) * hypergeom([-n, 2*n + 1], [n + 2], 5/4). - Peter Luschny, Apr 12 2024
a(n) ~ sqrt(779 + 201*sqrt(41)) * (299 + 41^(3/2))^n / (sqrt(41*Pi) * n^(3/2) * 2^(3*n + 5/2) * 5^(n+1)). - Vaclav Kotesovec, Apr 12 2024
From Peter Bala, Sep 08 2024: (Start)
G.f. A(x) = 1 + series_reversion( x/((1 + 5*x)*(1 + x)^2) ).
A(x) = 1 + x*(5*A(x)^3 - 4*A(x)^2). (End)
Extensions
More terms from Jun Yan, Apr 12 2024
Comments