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A078481
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Expansion of (1 - x - x^2 - sqrt(1 - 2*x - 5*x^2 - 2*x^3 + x^4)) / (2*x + 2*x^2).
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6
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0, 1, 1, 3, 7, 19, 53, 153, 453, 1367, 4191, 13015, 40857, 129441, 413337, 1328971, 4298727, 13978971, 45673981, 149867513, 493638797, 1631616239, 5410015615, 17990076527, 59981630321, 200476419713, 671564145137, 2254338511507, 7582179238151, 25547868961315
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OFFSET
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0,4
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COMMENTS
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Number of irreducible stack sortable permutations of degree n.
Also number of Dyck paths of semilength n with no UDUD. Example: a(3)=3 because the only Dyck paths of semilength 3 with no UDUD in them are: UDUUDD, UUDDUD and UUUDDD (the nonqualifying ones being UUDUDD and UDUDUD). - Emeric Deutsch, Jan 27 2003
The sequence 1,1,1,3,7,19,... has general term sum{k=0..n, C(n+k,2k)*(-1)^(n-k)*A006318(k)} and g.f. given by
1/(1+x-2x/(1+x-x/(1+x-2x/(1+x-x/(1+x-2x/(1+x-x/(1-..... (continued fraction). (End)
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LINKS
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Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
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FORMULA
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G.f.: (1 - x - x^2 - (1 - 2*x - 5*x^2 - 2*x^3 + x^4)^(1/2)) / (2*x + 2*x^2) = -1 + 2*(1 + x) / (1 + x + x^2 + sqrt((1 - x + x^2)^2 - 8*x^2)).
G.f. A(x) satisfies A(x) = x + (x + x^2) * (A(x) + A(x)^2). - Michael Somos, Sep 08 2005
Given g.f. A(x), then series reversion of B(x) = x + x*A(x) is -B(-x). - Michael Somos, Sep 08 2005
Given g.f. A(x), then B(x) = x + x*A(x) satisfies 0 = f(x, B(x)) where f(u, v) = u^2*(v + v^2) + u*(1 + v + v^2) - v. - Michael Somos, Sep 08 2005
Given g.f. A(x), then B(x) = x + x*A(x) satisfies B(x) = x + C(x*B(x)) where C(x) is g.f. of A006318 with offset 1. - Michael Somos, Sep 08 2005
D-finite with recurrence: (n+1)*a(n) +(-n+2)*a(n-1) +(-7*n+11)*a(n-2) +(-7*n+17)*a(n-3) +(-n+2)*a(n-4) +(n-5)*a(n-5)=0. - R. J. Mathar, Nov 26 2012
a(n) = sum(k=0..n, ((sum(i=0..n-k, binomial(k+1,n-k-i)*binomial(k+i,k)))*binomial(n-k-2,k))/(k+1)), n>0, a(0)=0. - Vladimir Kruchinin, Nov 22 2014.
a(n) ~ sqrt(2 - 1/sqrt(2) + sqrt(7*(4*sqrt(2)-5)/2)) * ((1 + 2*sqrt(2) + sqrt(5 + 4*sqrt(2)))/2)^n / (2 * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Jan 27 2015
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EXAMPLE
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x + x^2 + 3*x^3 + 7*x^4 + 19*x^5 + 53*x^6 + 153*x^7 + 453*x^8 + 1367*x^9 + ...
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MATHEMATICA
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CoefficientList[Series[(1 - x - x^2 - (1 - 2*x - 5*x^2 - 2*x^3 + x^4)^(1/2)) / (2*x + 2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 27 2015 *)
CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 2 x - 5 x^2 - 2 x^3 + x^4]) / (2 x + 2 x^2), {x, 0, 33}], x] (* Vincenzo Librandi, May 27 2016 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, polcoeff( -1 + 2*(1 + x) / (1 + x + x^2 + sqrt((1 - x + x^2)^2 - 8*x^2 + x*O(x^n))), n))} /* Michael Somos, Sep 08 2005 */
(Maxima) a(n):=if n=0 then 0 else sum(((sum(binomial(k+1, n-k-i)*binomial(k+i, k), i, 0, n-k))*binomial(n-k-2, k))/(k+1), k, 0, n); /* Vladimir Kruchinin, Nov 22 2014 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Replaced definition with g.f. given by Atkinson and Still (2002). - N. J. A. Sloane, May 24 2016
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STATUS
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approved
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