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A132889
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Sum of the lengths of the longest increasing subsequence over all 321-avoiding permutations of [n].
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2
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1, 3, 11, 39, 144, 530, 1987, 7455, 28268, 107334, 410354, 1570954, 6042984, 23273172, 89948835, 348000975, 1350028020, 5241881150, 20396787070, 79426533758, 309829067496, 1209384071532, 4727454837846, 18490127530394
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OFFSET
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1,2
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COMMENTS
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Apparently: Summed over all Dyck (n+1)-paths, the number of ascents of length 1 to the right of the midpoint (ordinate x=n+1). - David Scambler, Aug 08 2012
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LINKS
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FORMULA
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a(n) = Sum_{k=floor((n+1)/2)..n} k*(2*k-n+1)^2*binomial(n+1,n-k)^2 / (n+1)^2.
a(n) = Sum_{k=1..n} k*A126217(n,k).
D-finite Recurrence: n^2*(n+1)*(4*n^3 - 13*n^2 - 3*n + 24)*a(n) = 2*n*(8*n^5 - 14*n^4 - 67*n^3 + 152*n^2 - 67*n + 12)*a(n-1) + 8*(8*n^6 - 66*n^5 + 156*n^4 + 6*n^3 - 461*n^2 + 543*n - 180)*a(n-2) - 32*(n-3)*(n-2)*(2*n - 5)*(4*n^3 - n^2 - 17*n + 12)*a(n-3). - Vaclav Kotesovec, Mar 20 2014
a(n) ~ 2^(2*n-1)/sqrt(Pi*n) * (1 + 2/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
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EXAMPLE
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a(3)=11 because in the 321-avoiding permutations of 123, namely 123,132,312,213 and 231, the lengths of the longest increasing subsequences are 3,2,2,2 and 2, respectively.
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MAPLE
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a:=proc(n) options operator, arrow: (sum(k*(2*k-n+1)^2*binomial(n+1, n-k)^2, k =floor((1/2)*n+1/2)..n))/(n+1)^2 end proc: seq(a(n), n=1..25);
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MATHEMATICA
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Table[Sum[k*(2*k-n+1)^2*Binomial[n+1, n-k]^2, {k, Floor[(n+1)/2], n}]/(n+1)^2, {n, 1, 20}] (* Vaclav Kotesovec, Mar 20 2014 *)
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PROG
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(PARI) for(n=1, 25, print1(sum(k=floor((n+1)/2), n, k*(2*k-n+1)^2 * binomial(n+1, n-k)^2/(n+1)^2), ", ")) \\ G. C. Greubel, May 25 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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