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A098746
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Number of permutations of [1..n] which avoid 4231 and 42513.
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6
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1, 1, 2, 6, 23, 102, 495, 2549, 13682, 75714, 428882, 2474573, 14492346, 85926361, 514763279, 3111119358, 18946375767, 116147683902, 716179441293, 4438862153246, 27638747494178, 172805469880497, 1084462349973559, 6828717036765622, 43132158190994223, 273204023401012901
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OFFSET
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0,3
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COMMENTS
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(a(n))_{n>=1} is the INVERT transform of (u(n))_{n>=1}:=(1,1,3,12,55,273,...), the ternary numbers A001764. - David Callan, Nov 21 2011
a(n) = number of Dyck paths of semilength 2n for which all descents are of even length (counted by A001764) with no valley vertices at height 1. For example, a(2)=2 counts UUUUDDDD, UUDDUUDD. - David Callan, Nov 21 2011
Conjecture: a(n) is the number of permutations of [1..n] which avoid 1342 and 13254. - Alexander Burstein, Oct 19 2017
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LINKS
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FORMULA
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G.f.: 1 + Sum_{n>=1} (t^n*Sum_{k=0..n} ((n-k)*binomial(2*k+n,k)/(2*k+n))).
G.f.: sqrt(3)/(sqrt(3)-2*sqrt(x)*sin(asin(3*sqrt(3x)/2)/3)). - Paul Barry, Dec 15 2006
Let M = the production matrix:
1, 1;
1, 2, 1;
1, 3, 2, 1;
1, 4, 3, 2, 1;
1, 5, 4, 3, 2, 1;
...
a(n) is the upper left term in M^n, with sum of top row terms = a(n+1). Example: top row of M^3 = (6, 11, 5, 1), where a(3) = 6 and a(4) = 23 = (6 + 11 + 5 + 1). (End)
a(n) ~ 3^(3*n+3/2) / (49 * sqrt(Pi) * 4^n * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014
Conjecture: 2*(2*n-1)*(n-1)*a(n) +3*(11*n^2-67*n+76)*a(n-1) +3*(-155*n^2+931*n-1356)*a(n-2) +(469*n^2-2799*n+4070)*a(n-3) -48*(3*n-8)*(3*n-10)*a(n-4)=0. - R. J. Mathar, May 30 2014
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MAPLE
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1+add( t^n * add( (n-l)*binomial(2*l+n, l)/(2*l+n), l=0..n ), n=1..30);
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MATHEMATICA
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Flatten[{1, Table[Sum[(n-j)*Binomial[2*j+n, j]/(2*j+n), {j, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Mar 17 2014 *)
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PROG
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(PARI) a(n) = {my(k = 1); if(n > 0, k = sum(j = 0, n, (n-j)*binomial(2*j+n, j)/(2*j+n))); k; } \\ Jinyuan Wang, Aug 03 2019
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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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