The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A078481 Expansion of (1 - x - x^2 - sqrt(1 - 2*x - 5*x^2 - 2*x^3 + x^4)) / (2*x + 2*x^2). 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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 From Paul Barry, Jan 29 2009: (Start) 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) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 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. Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019). M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Preprint, 2002. M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Discrete Math., 259 (2002), 19-36. J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). See Table 4. Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7. Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5. A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. FORMULA 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 EXAMPLE x + x^2 + 3*x^3 + 7*x^4 + 19*x^5 + 53*x^6 + 153*x^7 + 453*x^8 + 1367*x^9 + ... MATHEMATICA 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 *) PROG (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 */ CROSSREFS Cf. A006318, A094507. Sequence in context: A026299 A183117 A183124 * A183122 A104522 A351633 Adjacent sequences: A078478 A078479 A078480 * A078482 A078483 A078484 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 04 2003 EXTENSIONS Replaced definition with g.f. given by Atkinson and Still (2002). - N. J. A. Sloane, May 24 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified February 21 03:13 EST 2024. Contains 370219 sequences. (Running on oeis4.)