This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A105633 Row sums of triangle A105632. 13
 1, 2, 4, 9, 22, 57, 154, 429, 1223, 3550, 10455, 31160, 93802, 284789, 871008, 2681019, 8298933, 25817396, 80674902, 253106837, 796968056, 2517706037, 7977573203, 25347126630, 80738862085, 257778971504, 824798533933 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Binomial transform of A007477. INVERT transform of A082582. First differences give A086581 and A025242 (offset 1). Is this sequence equal to A057580? a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU = the number of binary trees without zigzag (i.e., with no node with a father, with a right son and with no left son). This sequence is the first column of the triangle A116424. E.g., a(2) = 4 because there exist four Dyck paths of semilength 3 that avoid UUDU: UDUDUD, UDUUDD, UUDDUD, UUUDDD, as well as four Dyck paths of semilength 3 that avoid UDUU: UDUDUD, UUDUDD, UUDDUD, UUUDDD. - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006 The sequence beginning 1,1,2,4,9,... gives the diagonal sums of A130749, and has g.f. 1/(1-x-x^2/(1-x/(1-x-x^2/(1-x/(1-x-x^2/(1-... (continued fraction); and general term Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)*A090181(j,k). Its Hankel transform is A099443(n+1). - Paul Barry, Jun 30 2009 The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al. - Kellen Myers, Jun 15 2015 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018. Maciej Bendkowski, Quantitative aspects and generation of random lambda and combinatory logic terms, Ph.D. Thesis, Jagiellonian University, Kraków, 2017. Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, Combinatorics of λ-terms: a natural approach, arXiv:1609.08106 [cs.LO], 2016. Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, A Natural Counting of Lambda Terms, arXiv preprint arXiv:1506.02367 [cs.LO], 2015. Maciej Bendkowski, K Grygiel, P Tarau, Random generation of closed simply-typed lambda-terms: a synergy between logic programming and Boltzmann samplers, arXiv preprint arXiv:1612.07682 [cs.LO], 2016-2017. Juan B. Gil, Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018. K. Grygiel, P. Lescanne, A natural counting of lambda terms, Preprint 2015. Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221. Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5. A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741. FORMULA G.f.: A(x) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x)))/(2*x^2). a(n) = 2*a(n-1) + Sum_{i=1..n-2} a(i)*(a(n-1-i) - a(n-2-i)). a(n) = Sum_{i=0..floor(n/2)} (-1)^i * binomial(n+1-i,i) * binomial(2*(n+1)-3*i, n-2*i) /(n+1-i). - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006 G.f.: (1/(1-x)^2)c(x^2/(1-x)^3), where c(x) is the g.f. of A000108. - Paul Barry, May 22 2009 1/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Jun 30 2009 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)(0^(j+k)+(1/(j+0^j))*binomial(j,k)*binomial(j,k+1)). - Paul Barry, Jun 30 2009 G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*(1-x)*A(x)). - Paul D. Hanna, Sep 12 2012 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * (1-x)^k ). - Paul D. Hanna, Sep 12 2012 Conjecture: (n+2)*a(n) + (-4*n-3)*a(n-1) + (2*n+1)*a(n-2) + a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012 The conjecture is true, since by holonomic transformation, it can be computed formally using GFUN, associated with the equation: x^3 + x^2 - 2x + (x^3 + 3 x^2 -3x +1) A(x) + (x^5 + 2x^3 -4 x^2 + x) A'(x) = 0. - Pierre Lescanne, Jun 30 2015 G.f.: (1 - 1/(G(0)-x))/x^2 where G(k) =  1 + x/(1 + x/(x^2 - 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 16 2012 a(n) ~ 2^(n/3-1/6) * 3^(n+2) * (13+3*sqrt(33))^((n+1)/3) * sqrt(4*(2879 + 561*sqrt(33))^(1/3) + 8*(7822 + 1362*sqrt(33))^(1/3) - 91 - 21*sqrt(33)) / (((26+6*sqrt(33))^(2/3) - (26+6*sqrt(33))^(1/3) - 8)^(n+3/2) * (4*(26+6*sqrt(33))^(1/3) - (26+6*sqrt(33))^(2/3) + 8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Mar 13 2014 a(n) = Sum_{i=0..n+1} (-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4). - Peter Luschny, May 03 2018 EXAMPLE G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 22*x^4 + 57*x^5 + 154*x^6 + 429*x^7 + ... with A(x)^2 = 1 + 4*x + 12*x^2 + 34*x^3 + 96*x^4 + 274*x^5 + 793*x^6 + ... where A(x) = 1 + x*(2-x)*A(x) + x^2*(1-x)*A(x)^2. The logarithm of the g.f. begins: log(A(x)) = (1 + (1-x))*x + (1 + 2^2*(1-x) + (1-x)^2)*x^2/2 + (1 + 3^2*(1-x) + 3^2*(1-x)^2 + (1-x)^3)*x^3/3 + (1 + 4^2*(1-x) + 6^2*(1-x)^2 + 4^2*(1-x)^3 + (1-x)^4)*x^4/4 + (1 + 5^2*(1-x) + 10^2*(1-x)^2 + 10^2*(1-x)^3 + 5^2*(1-x)^4 + (1-x)^5)*x^5/5 + ... Explicitly, log(A(x)) = 2*x + 4*x^2/2 + 11*x^3/3 + 32*x^4/4 + 97*x^5/5 + 301*x^6/6 + 947*x^7/7 + 3008*x^8/8 + 9623*x^9/9 + 30959*x^10/10 + ... MAPLE a := n -> add((-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4), i=0..n+1): seq(simplify(a(n)), n=0..26); # Peter Luschny, May 03 2018 MATHEMATICA CoefficientList[Series[(1 - x - Sqrt[(1 - x)^2 - 4 x^2/(1 - x)])/(2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *) PROG (PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(2/(1-X)/(1-X+sqrt((1-X)^2-4*X^2/(1-X))), n, x)} (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=0, m, binomial(m, k)^2*(1-x)^(m-k) + x*O(x^n)))), n)} \\ Paul D. Hanna, Sep 12 2012 CROSSREFS Cf. A105632, A057580, A116424, A216604. See also A258973. Sequence in context: A130018 A322504 A099754 * A287709 A196161 A249561 Adjacent sequences:  A105630 A105631 A105632 * A105634 A105635 A105636 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 17 2005 EXTENSIONS More terms from I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006 STATUS approved

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

Last modified September 19 08:43 EDT 2019. Contains 327189 sequences. (Running on oeis4.)