This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A234797 E.g.f. satisfies: A'(x) = 1 + A(x) + 2*A(x)^2, where A(0)=0. 3
 1, 1, 5, 17, 109, 649, 5285, 44513, 448861, 4836601, 58743125, 766520657, 10939702669, 167136559849, 2746173173765, 48016925121473, 893361709338301, 17582667488919001, 365487998075525045, 7994319232001122097, 183644125043688405229, 4418905413530661307849 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n) = number of increasing ordered trees on the vertex set {1,2,...,n}, rooted at 1, in which all outdegrees are <= 2 and the vertices of degree 2 are colored either white or black. An example is given below. - Peter Bala, Sep 13 2015 LINKS F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of increasing trees, preprint. F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of increasing trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48. FORMULA E.g.f.: Series_Reversion( Integral 1/(1 + x + 2*x^2) dx ). E.g.f.: (sqrt(7)*tan(sqrt(7)*x/2 + arctan(1/sqrt(7)))-1)/4. - Vaclav Kotesovec, Jan 13 2014 a(n) ~ n! * 1/2*(sqrt(7)/(Pi - 2*arctan(1/sqrt(7))))^(n+1). - Vaclav Kotesovec, Jan 13 2014 O.g.f.: A(x) = x/(1-x - 2*1*2*x^2/(1-2*x - 2*2*3*x^2/(1-3*x - 2*3*4*x^2/(1-... -n*x - 2*n*(n+1)*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jun 12 2014 Let f(x) = 1 + x + 2*x^2. Then a(n+1) = (f(x)*d/dx)^n f(x) evaluated at x = 0. - Peter Bala, Sep 13 2015 G.f.: 1/T(0), where m=4; u=x; T(k)= 1 - u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/( 1 - u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2015 a(n) = 2^n*A(n, 1/2) where A(n,x) are the AndrĂ© polynomials. - Peter Luschny, May 05 2016 EXAMPLE E.g.f.: A(x) = x + x^2/2! + 5*x^3/3! + 17*x^4/4! + 109*x^5/5! + 649*x^6/6! +... Related series. A(x)^2 = 2*x^2/2! + 6*x^3/3! + 46*x^4/4! + 270*x^5/5! + 2318*x^6/6! +... a(4) = 17. The 17 plane (ordered) increasing unary-binary trees on 4 vertices are shown below. A * indicates the vertex of outdegree 2 may be colored either white or black. ................................................................... ..4................................................................ ..|................................................................ ..3..........4...4...............4...4...............3...3......... ..|........./.....\............./.....\............./.....\........ ..2....2...3.......3...2...3...2.......2...3...4...2.......2...4... ..|.....\./.........\./.....\./.........\./.....\./.........\./.... ..1......1*..........1*......1*..........1*......1*..........1*.... ................................................................... ..3...4...4...3.................................................... ...\./.....\./..................................................... ....2*......2*...................................................... ....|.......|...................................................... ....1.......1...................................................... ................................................................... - Peter Bala, Sep 13 2015 MATHEMATICA Rest[FullSimplify[CoefficientList[Series[(Sqrt[7]*Tan[Sqrt[7]*x/2 + ArcTan[1/Sqrt[7]]]-1)/4, {x, 0, 20}], x] * Range[0, 20]!]] (* Vaclav Kotesovec, Jan 13 2014 *) nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - x*(2*k+1) - 4*x^2*(k+1)*(2*k+1)/( 1 - x*(2*k+2) - 4*x^2*(k+1)*(2*k+3)/g[k+1] ); CoefficientList[Series[1/g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 15 2015, after Sergei N. Gladkovskii *) PROG (PARI) {a(n)=local(A=x); for(i=1, n, A=intformal(1+A+2*A^2 +x*O(x^n))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) (PARI) {a(n)=local(A=serreverse(intformal(1/(1+x+2*x^2 +x*O(x^n))))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) (Sage) @CachedFunction def c(n, k) :     if n==k: return 1     if k<1 or k>n: return 0     return ((n-k)//2+1)*c(n-1, k-1)+k*c(n-1, k+1) def A234797(n):     return add(c(n, k)*2^(n-k) for k in (0..n)) [A234797(n) for n in (1..22)] # Peter Luschny, Jun 10 2014 CROSSREFS Cf. A094503. Sequence in context: A096178 A084167 A267143 * A062586 A301641 A067710 Adjacent sequences:  A234794 A234795 A234796 * A234798 A234799 A234800 KEYWORD nonn,easy AUTHOR Paul D. Hanna, Jan 09 2014 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 October 21 06:39 EDT 2019. Contains 328292 sequences. (Running on oeis4.)