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A243676
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Number of hypoplactic classes of 4-parking functions of length n.
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9
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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Table of n, a(n) for n=0..5.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 23.
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FORMULA
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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
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CROSSREFS
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Appears to equal A260332(n) 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 A356240 A157551
Adjacent sequences: A243673 A243674 A243675 * A243677 A243678 A243679
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KEYWORD
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nonn,more
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AUTHOR
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N. J. A. Sloane, Jun 14 2014
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EXTENSIONS
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Added a(0)=1. - N. J. A. Sloane, Mar 28 2021
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STATUS
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approved
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