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A243667
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Number of Sylvester classes of 4-packed words of degree n.
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4
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1, 1, 6, 50, 484, 5105, 56928, 660112, 7878940, 96159476, 1194532794, 15053992178, 191993403476, 2473358617150, 32137897641232, 420698195672700, 5542894551818268, 73447821835338348, 978178443083177880, 13086377223959022952, 175785879063917657688
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OFFSET
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0,3
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COMMENTS
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See Novelli-Thibon (2014) for precise definition.
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..865
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 Eq. (185), p. 47 and Fig. 17.
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FORMULA
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Novelli-Thibon give an explicit formula in Eq. (182).
From Seiichi Manyama, Jul 26 2020: (Start)
G.f. A(x) satisfies: A(x) = 1 - x * A(x)^4 * (1 - 2 * A(x)).
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(4*n+k+1,n)/(4*n+k+1).
a(n) = ( (-1)^n / (4*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+1,k) * binomial(5*n-k,n-k). (End)
a(n) ~ 2^(9*n - 15) * sqrt(436289 + 2793997/sqrt(41)) / (sqrt(Pi) * n^(3/2) * (29701 - 4633*sqrt(41))^(n - 1/2)). - Vaclav Kotesovec, Jul 31 2021
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MATHEMATICA
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P[n_, m_, x_] := 1/(m n + 1) Sum[Binomial[m n + 1, k] Binomial[(m + 1) n - k, n - k] (1 - x)^k x^(n - k), {k, 0, n}];
a[n_] := P[n, 4, 2];
a /@ Range[20] (* Jean-François Alcover, Jan 28 2020 *)
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PROG
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(PARI) {a(n) = local(A=1+x*O(x^n)); for(i=0, n, A=1-x*A^4*(1-2*A)); polcoeff(A, n)} \\ Seiichi Manyama, Jul 26 2020
(PARI) {a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(4*n+k+1, n)/(4*n+k+1))} \\ Seiichi Manyama, Jul 26 2020
(PARI) {a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(4*n+1, k)*binomial(5*n-k, n-k))/(4*n+1)} \\ Seiichi Manyama, Jul 26 2020
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CROSSREFS
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Column k=4 of A336573.
Cf. A243668, A336572.
Sequence in context: A180910 A199680 A039742 * A303562 A125558 A355421
Adjacent sequences: A243664 A243665 A243666 * A243668 A243669 A243670
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KEYWORD
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nonn
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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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More terms from Jean-François Alcover, Jan 28 2020
a(0)=1 prepended by Seiichi Manyama, Jul 25 2020
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STATUS
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approved
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