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%I #57 Apr 14 2024 04:49:33
%S 1,5,33,249,2033,17485,156033,1431281,13412193,127840085,1235575201,
%T 12080678505,119276490193,1187542872989,11909326179841,
%U 120191310803937,1219780566014657,12440630635406245,127446349676475425,1310820823328281561,13530833791486094769
%N Small 3-Schroeder numbers: a(n) = A027307(n+1)/2.
%C Series reversion of x*(Sum_{k>=0} a(k)(-x^2)^k) is Sum_{k odd} C(k)x^k where C() is Catalan numbers A000108.
%C Series reversion of x*(Sum_{k>=0} a(k)(-x)^k) is A000337(x). (_Michael Somos_)
%C This sequence should really have started with a(0)=1, a(1)=1, a(2)=5, a(3)=33, ..., but the present offset is too well-established. - _N. J. A. Sloane_, Mar 28 2021
%C This is the number of hypoplactic classes of 2-parking functions of size n+1. - _Jun Yan_, Apr 13 2024
%D 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.
%H Alois P. Heinz, <a href="/A034015/b034015.txt">Table of n, a(n) for n = 0..500</a>
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
%H Jun Yan, <a href="http://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv preprint arXiv:2404.07958 [math.CO], 2024. See Theorem 4.4.
%F a(n) = Sum_{i=0..n} Sum_{j=0..i} (-2)^(n-i)*binomial(i,j)*binomial(2i+j, n)*binomial(n+1,i)/(n+1) (conjectured). - _Michael D. Weiner_, May 25 2017
%F Yang & Jiang (2021) give an explicit formula for a(n) in Theorems 2.4 and 2.9. - _N. J. A. Sloane_, Mar 28 2021 [This formula is: a(n) = (1/(n + 1)) * Sum_{k=1..n+1} binomial(2*n + 2, k - 1) * binomial(n + 1, k)*2^(k - 1). - _Jun Yan_, Apr 13 2024]
%F a(n) = hypergeom([-n, -2*(n + 1)], [2], 2). - _Peter Luschny_, Nov 08 2021
%F a(n) ~ phi^(5*n + 6) / (4 * 5^(1/4) * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Nov 08 2021
%F D-finite with recurrence +2*(2*n+3)*(n+1)*a(n) +(-46*n^2-43*n-9)*a(n-1) +3*(6*n^2-14*n+7)*a(n-2) +(2*n-3)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Aug 01 2022
%F Let D(n) be the set of 2-Dyck paths that have n up-steps of size 2 and 2n 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+1)}2^(peak(d) - 1). - _Jun Yan_, Apr 13 2024
%p a:= proc(n) option remember; `if`(n<2, 4*n+1,
%p ((110*n^3+66*n^2-17*n-9) *a(n-1)
%p +(n-1)*(2*n-1)*(5*n+3) *a(n-2)) /
%p ((2*n+3)*(5*n-2)*(n+1)))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jun 22 2014
%t a[n_] := If[n<0, 0, Sum[2^i*Binomial[2*n+2, i]*Binomial[n+1, i+1]/(n+1), {i, 0, n}]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Oct 13 2014, after PARI *)
%t a[n_] := Hypergeometric2F1[-n, -2 (n + 1), 2, 2];
%t Table[a[n], {n, 0, 20}] (* _Peter Luschny_, Nov 08 2021 *)
%o (PARI) a(n)=if(n<0,0,sum(i=0,n,2^i*binomial(2*n+2,i)*binomial(n+1,i+1))/(n+1))
%Y Part of a family indexed by m: m=2 (A001003), m=3 is this sequence, m=4 is A243675, ....
%Y 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
%Y Apart from the first term, this is A027307/2. - _N. J. A. Sloane_, Mar 28 2021
%K nonn,changed
%O 0,2
%A _N. J. A. Sloane_
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