cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A108529 Number of asymmetric mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 1, 2, 5, 16, 51, 177, 621, 2246, 8245, 30783, 116257, 443945, 1710255, 6640939, 25961690, 102105115, 403701135, 1603721999, 6397931901, 25621989760, 102965680728, 415091909292, 1678226164646, 6803121058354, 27645628327636
Offset: 1

Views

Author

Christian G. Bower, Jun 07 2005

Keywords

Comments

A generator is a leaf or a node with just one child.
Here CHK(A(x)) = 1 - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)), i.e., the constant 1 is included in the definition of the CHK transform. For other sequences that involve the CHK transform, the 1 is sometimes dropped; e.g., see sequence A032171. We have CHK(A(x)) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 86*x^6 + 303*x^7 + 1065*x^8 + 3871*x^9 + ... - Petros Hadjicostas, Dec 05 2017

Links

Crossrefs

Cf. A108521, A108522, A108523, A108524, A108525, A108527, A108528, A032171.

Programs

  • PARI
    CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=x); for(n=2, n, p += x^n*polcoef(x*p + CHK(p, n), n)); Vecrev(p/x)} \\ Andrew Howroyd, Aug 31 2018

Formula

G.f. satisfies: (2-x)*A(x) = x - 1 + CHK(A(x)).
From Petros Hadjicostas, Dec 05 2017: (Start)
a(n) = (1/2)*(a(n-1) + (1/n)*Sum_{d|n} mu(d)*c(n/d)) for n>=2, where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) and a(1) = c(1) = 1.
The g.f. satisfies (2-x)*A(x) = x - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)). (This is just a rephrasing of C. Bower's equation above.)
The auxiliary sequence (c(n): n>=1} has g.f. C(x) = Sum_{n>=1} c(n)*x^n = x*(dA/dx)/(1-A(x)) = x + 3*x^2 + 10*x^3 + 35*x^4 + 136*x^5 + 528*x^6 + 2122*x^7 + ...
(End)

A108524 Number of ordered rooted trees with n generators.

Original entry on oeis.org

1, 2, 7, 32, 166, 926, 5419, 32816, 203902, 1292612, 8327254, 54358280, 358769152, 2390130038, 16051344307, 108548774240, 738563388214, 5052324028508, 34727816264050, 239733805643552, 1661351898336676, 11553558997057772, 80603609263563262, 563972937201926432
Offset: 1

Views

Author

Christian G. Bower, Jun 07 2005

Keywords

Comments

A generator is a leaf or a node with just one child.
The Hankel transform of this sequence is 3^C(n+1,2). The Hankel transform of this sequence with 1 prepended (1,1,2,7,...) is 3^C(n,2). - Paul Barry, Jan 26 2011
a(n) is the number of Schroder paths of semilength n-1 in which the (2,0)-steps that are not on the horizontal axis come in 2 colors. Example: a(3)=7 because we have HH, UDUD, UUDD, HUD, UDH, UBD, and URD, where U=(1,1), H=(2,0), D=(1,-1), while B and R are, respectively, blue and red (2,0)-steps. - Emeric Deutsch, May 02 2011
Also the compositional inverse of A327767. - Peter Luschny, Oct 08 2022

Links

Crossrefs

Cf. A108521-A108529, A108525, A000108, A001003, A327767.

Programs

  • Maple
    # Using function CompInv from A357588.
CompInv(24, n -> [1, -2][irem(n-1, 2) + 1]); # Peter Luschny, Oct 08 2022
  • Mathematica
    Rest[CoefficientList[Series[(Sqrt[4*x^2-8*x+1]-1)/(2*x-4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 18 2012 *)
  • Maxima
    a(n):=sum((i*binomial(n+1,i)*sum((-1)^j*2^(n-j)*binomial(n,j)*binomial(2*n-j-i-1,n-1),j,0,n-i))/2^i,i,1,n+1)/(n*(n+1)); /* Vladimir Kruchinin, May 10 2011 */

Formula

G.f.: (sqrt(4*x^2-8*x+1) - 1)/(2*x-4).
G.f.: 1/(1-x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction). - Paul Barry, Feb 10 2009
a(n) = sum(i=1..n+1, (i*C(n+1,i)*sum(j=0..n-i, (-1)^j*2^(n-j)*C(n,j)*C(2*n-j-i-1,n-1)))/2^i)/(n*(n+1)). - Vladimir Kruchinin, May 10 2011
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) is upper left term in the following infinite square production matrix:
1, 1, 0, 0, 0, ...
1, 1, 1, 0, 0, ...
3, 3, 1, 1, 0, ...
9, 9, 3, 1, 1, ...
...
where columns are (1, 1, 3, 9, 27, 81, ...) prefaced with (0,0,1,2,3,...) zeros. (End)
Conjecture: 2*n*a(n) +(24-17*n)*a(n-1) +4*(4*n-9)*a(n-2) +4*(3-n)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
G.f.: A(x)=(sqrt(4*x^2-8*x+1) - 1)/x/(2*x-4) = 1/(G(0)-x); G(k) = 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012
a(n) ~ 3^(1/4)*(3^(3/2)-5)*(4+2*sqrt(3))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012
From Peter Bala, Mar 13 2015: (Start)
The o.g.f. A(x) satisfies the differential equation (2 - 17*x + 16*x^2 - 4*x^3)A'(x) + (7 - 4*x)*A(x) = 2 - 2*x. Mathar's conjectural recurrence above follows from this.
The o.g.f. A(x) is the series reversion of the rational function x*(1 - 2*x)/(1 - x^2). (End)

A108528 Number of increasing mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 2, 10, 92, 1216, 20792, 435520, 10793792, 308874016, 10021509632, 363509706880, 14576530558592, 640275236943616, 30573223563625472, 1576805482203235840, 87353392124392020992, 5173324070004374358016, 326160898887563325581312, 21810458629345555407462400
Offset: 1

Views

Author

Christian G. Bower, Jun 07 2005

Keywords

Comments

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.

Links

Crossrefs

Cf. A108521, A108522, A108523, A108524, A108525, A108526, A108527, A108528, A108529, A029768.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[Log[(1+x)*Sqrt[1-x^2]], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(log((1+x)*sqrt(1-x^2+O(x^(n+2))))),n)} \\ Paul D. Hanna, Sep 11 2010

Formula

E.g.f. satisfies 2*A(x) = x - 1 + A'(x) - log(1-A(x)).
From Paul D. Hanna, Sep 11 2010: (Start)
E.g.f. satisfies: (1+A(x))*sqrt(1-A(x)^2) = exp(x).
E.g.f.: A(x) = Series_Reversion[ log((1+x)*sqrt(1-x^2)) ]. (End)
a(n) ~ 2^(n-2) * sqrt(3) * n^(n-1) / (exp(n) * (log(27/16))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014
Showing 1-3 of 3 results.

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