A108531 -id:A108531 - cp's OEIS frontend

A052716 Expansion of e.g.f. (x + 1 - sqrt(1-6*x+x^2))/2.

0, 2, 4, 36, 528, 10800, 283680, 9102240, 345058560, 15090727680, 747888422400, 41422381862400, 2535569103513600, 169983582318950400, 12386182292118835200, 974723523832041984000, 82385641026424479744000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

With a(n)=1, also number of labeled mobiles with n nodes and 2-colored internal (non-leaf) nodes - Christian G. Bower, Jun 07 2005

G. C. Greubel, Table of n, a(n) for n = 0..345
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 672
Index entries for sequences related to mobiles

Cf. A006318, A108531.

[n le 1 select 1-(-1)^n else Factorial(n)*(&+[Catalan(k)*Binomial(n+k-1, n-k-1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, May 28 2022

spec := [S,{C=Union(B,Z),B=Prod(S,C),S=Union(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(x+1-Sqrt[1-6x+x^2])/2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Apr 19 2020 *)

[bool(n==1)+factorial(n)*sum(binomial(n+k-1, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(2)=4, a(1)=2, (n^2-1)*a(n) = (3+6*n)*a(n+1) - a(n+2).

a(n) = n!*A006318(n-1), n>=2. - R. J. Mathar, Oct 26 2013

A108530 Number of rooted identity trees with n internal (non-leaf) nodes.

Original entry on oeis.org

1, 1, 2, 4, 12, 34, 110, 364, 1248, 4356, 15520, 56022, 204726, 755472, 2812004, 10543718, 39791070, 151022006, 576090250, 2207493080, 8493196536, 32797115398, 127071214442, 493831241234, 1924504466246, 7519182311366, 29447430754182, 115577336981932

Offset: 0

Christian G. Bower, Jun 07 2005

Also for n>0, rooted trees with n nodes and 2-colored internal nodes. Black nodes correspond to nodes with a leaf child; white nodes correspond to those without one.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A004111, A004113, A108531, A108532.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, 1, 2*b(n-1,n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 20 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<2, 1, 2*b[n-1, n-1]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Shifts left and halves under WEIGH transform.

a(n) ~ c * d^n / n^(3/2), where d = 4.1516890102085520777311008746639624... and c = 0.3329810927479684511418598248... - Vaclav Kotesovec, Feb 28 2014

A108532 Number of asymmetric mobiles (cycle rooted trees) with n nodes and 2-colored internal (non-leaf) nodes.

Original entry on oeis.org

1, 2, 4, 12, 38, 136, 490, 1852, 7108, 27880, 110892, 447060, 1821252, 7489732, 31045350, 129587996, 544228664, 2298008824, 9750218012, 41548438040, 177740526076, 763046178960, 3286318131646, 14195239150556, 61481540391722

Offset: 1

Christian G. Bower, Jun 07 2005

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for sequences related to mobiles
C. G. Bower, Transforms (2)

Cf. A032171, A108530, A108531.

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=O(1));for(i=1, n, p=1+2*CHK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left and halves under CHK transform.

A032202 Sequence (a(n): n >= 1) that shifts left 2 places under the "CIK" (necklace, indistinct, unlabeled) transform and satisfies a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 22, 41, 92, 193, 435, 963, 2215, 5051, 11754, 27375, 64381, 151898, 360661, 859149, 2055804, 4934428, 11883930, 28699336, 69497354, 168691424, 410399073, 1000486306, 2443761830, 5979742904, 14656709518

Offset: 1

Christian G. Bower

From Petros Hadjicostas, Dec 30 2018: (Start)
a(n+2) = (1/n)*Sum_{d|n} phi(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = a(2) = 1, c(1) = 1, and c(2) = 3.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+2)*x^n = -Sum_{n>=1} (phi(n)/n)*log(1-A(x^n)).
The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 46*x^5 + 117*x^6 + 281*x^7 + 707*x^8 + 1717*x^9 + 4288*x^10 + 10583*x^11 + 26401*x^12 + ...
(End)
The first two terms of the sequence must be specified. In general, if the sequence (b(n): n >= 1) is such that (b(n+2): n >= 1) = CIK((b(n): n >= 1)), then b(3) = b(1), b(4) = (1/2)*(b(1)^2 + 2*b(2) + b(1)), b(5) = (b(1)/3)*(b(1)^2 + 3*b(2) + 5), and so on. - Petros Hadjicostas, Jan 01 2019

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A032200, A032201, A108531.

m = 33; a[1] = a[2] = 1; A[_] = 0;
Do[A[x_] = x(a[1] + x a[2] - x Sum[EulerPhi[n] Log[1-A[x^n]]/n, {n, 1, m}]) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Sep 17 2019 *)

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=1+O(x));for(i=1, n\2, p=1+x+x*CIK(x*p, 2*i)); Vec(p+O(x^n))} \\ Andrew Howroyd, Jun 20 2018

Name modified by Petros Hadjicostas, Jan 01 2019

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A052716 Expansion of e.g.f. (x + 1 - sqrt(1-6*x+x^2))/2.

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A108530 Number of rooted identity trees with n internal (non-leaf) nodes.

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A108532 Number of asymmetric mobiles (cycle rooted trees) with n nodes and 2-colored internal (non-leaf) nodes.

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A032202 Sequence (a(n): n >= 1) that shifts left 2 places under the "CIK" (necklace, indistinct, unlabeled) transform and satisfies a(1) = a(2) = 1.

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