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A005802
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Number of permutations in S_n with longest increasing subsequence of length <= 3 (i.e., 1234-avoiding permutations); vexillary permutations (i.e., 2143-avoiding).
(Formerly M1666)
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30
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1, 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705, 167078577, 1148208090, 8026793118, 56963722223, 409687815151, 2981863943718, 21937062144834, 162958355218089, 1221225517285209, 9225729232653663, 70209849031116183, 537935616492552297
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OFFSET
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0,3
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COMMENTS
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Also the dimension of SL(3)-invariants in V^n tensor (V^*)^n, where V is the standard 3-dimensional representation of SL(3) and V^* is its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
Also the number of doubly-alternating permutations of length 2n with no four-term increasing subsequence (i.e., 1234-avoiding doubly-alternating permutations). The doubly-alternating permutations (counted by sequence A007999) are those permutations w such that both w and w^(-1) have descent set {2, 4, 6, ...}. - Joel B. Lewis, May 21 2009
Any permutation without an increasing subsequence of length 4 has a decreasing subsequence of length >= n/3, where n is the length of the sequence, by the Erdős-Szekeres theorem. - Charles R Greathouse IV, Sep 26 2012
Also the number of permutations of length n simultaneously avoiding patterns 1324 and 3416725 (or 1324 and 3612745). - Alexander Burstein, Jan 31 2014
For any integer n > 0, we have (n+2)^2*a(n) - n^2*a(n-1} = 4*A086618(n). - Zhi-Wei Sun, Nov 16 2017
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REFERENCES
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Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, pp. 65-82 of "A Century of Advancing Mathematics", ed. S. F. Kennedy et al., MAA Press 2015.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(e), p. 453.
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LINKS
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FORMULA
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a(n) = 2 * Sum_{k=0..n} binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2 + 2*k+1 - n - 2*k*n)/((k+1)^2 * (k+2) * (n-k+1)).
(4*n^2 - 2*n + 1)*(n + 2)^2*(n + 1)^2*a(n) = (44*n^3 - 14*n^2 - 11*n + 8)*n*(n + 1)^2*a(n - 1) - (76*n^4 + 42*n^3 - 49*n^2 - 24*n + 24)*(n - 1)^2*a(n - 2) + 9*(4*n^2 + 6*n + 3)*(n - 1)^2*(n - 2)^2*a(n - 3). - Vladeta Jovovic, Jul 16 2004
a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41) a(n + 1) - (9*n^2 + 18*n + 9) a(n). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
G.f.: (1+5*x-(1-9*x)^(3/4)*(1-x)^(1/4)*hypergeom([-1/4, 3/4],[1],64*x/((x-1)*(1-9*x)^3)))/(6*x^2). - Mark van Hoeij, Oct 25 2011
a(n) = Sum_{k=0..n} binomial(2k,k)*binomial(n+1,k+1)*binomial(n+2,k+1)/((n+1)^2*(n+2)). [Conway and Guttmann, Adv. Appl. Math. 64 (2015) 50]
a(n) = hypergeom([1/2, -1 - n, -n], [2, 2], 4) / (n+1). - Vaclav Kotesovec, Jun 07 2021
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MAPLE
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a:= n-> 2*add(binomial(2*k, k)*(binomial(n, k))^2*(3*k^2+2*k+1-n-2*k*n)/ (k+1)^2/(k+2)/(n-k+1), k=0..n);
A005802:=rsolve({a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41)*a(n + 1) - (9*n^2 + 18*n + 9)*a(n)}, a(n), makeproc): # Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
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MATHEMATICA
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a[n_] := 2Sum[Binomial[2k, k]Binomial[n, k]^2(3k^2+2k+1-n-2k*n)/((k+1)^2(k+2)(n-k+1)), {k, 0, n}]
(* Second program:*)
a[0] = a[1] = 1; a[n_] := a[n] = ((10*n^2+2*n-3)*a[n-1] + (-9*n^2+18*n-9)* a[n-2])/(n+2)^2; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 20 2017 *)
Table[HypergeometricPFQ[{1/2, -1 - n, -n}, {2, 2}, 4] / (n+1), {n, 0, 25}] (* Vaclav Kotesovec, Jun 07 2021 *)
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PROG
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(PARI) a(n)=2*sum(k=0, n, binomial(2*k, k)*binomial(n, k)^2*(3*k^2+2*k+1-n-2*k*n)/(k+1)^2/(k+2)/(n-k+1)) \\ Charles R Greathouse IV, Sep 26 2012
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
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STATUS
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approved
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