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A243668
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Number of Sylvester classes of 5-packed words of degree n.
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3
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1, 1, 7, 69, 793, 9946, 131993, 1822288, 25904165, 376601883, 5573626462, 83692267478, 1271883556731, 19525467196176, 302346907361688, 4716814859429384, 74065892877777885, 1169701519598447641, 18566836447453815317, 296053851068485920563, 4739945317989532651858
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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..812
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)^5 * (1 - 2 * A(x)).
a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(5*n+k+1,n)/(5*n+k+1).
a(n) = ( (-1)^n / (5*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+1,k) * binomial(6*n-k,n-k). (End)
a(n) ~ sqrt(27851068 + 7443921*sqrt(14)) * 5^(5*n - 13/2) / (sqrt(7*Pi) * n^(3/2) * 2^(2*(1 + n)) * (108007 - 28854*sqrt(14))^(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, 5, 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^5*(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(5*n+k+1, n)/(5*n+k+1))} \\ Seiichi Manyama, Jul 26 2020
(PARI) {a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(5*n+1, k)*binomial(6*n-k, n-k))/(5*n+1)} \\ Seiichi Manyama, Jul 26 2020
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CROSSREFS
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Column k=5 of A336573.
Cf. A243667.
Sequence in context: A180911 A084774 A025757 * A265033 A226270 A121351
Adjacent sequences: A243665 A243666 A243667 * A243669 A243670 A243671
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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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