A234797 E.g.f. satisfies: A'(x) = 1 + A(x) + 2*A(x)^2, where A(0)=0.
1, 1, 5, 17, 109, 649, 5285, 44513, 448861, 4836601, 58743125, 766520657, 10939702669, 167136559849, 2746173173765, 48016925121473, 893361709338301, 17582667488919001, 365487998075525045, 7994319232001122097, 183644125043688405229, 4418905413530661307849
Offset: 1
Examples
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
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.
Crossrefs
Cf. A094503.
Programs
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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 *) -
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
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 André polynomials. - Peter Luschny, May 05 2016
Comments