A055363 -id:A055363 - cp's OEIS frontend

A055370 Matrix inverse of triangle A055363(n+2,k).

1, -1, 1, 1, -2, 1, -1, 4, -4, 1, 0, -6, 10, -5, 1, 5, 1, -16, 16, -7, 1, -18, 38, -12, -26, 25, -8, 1, 27, -161, 197, -45, -44, 35, -10, 1, 67, 211, -580, 418, -82, -71, 47, -11, 1, -496, 1285, -629, -782, 839, -174, -91, 60, -13, 1, 496, -7131, 10638, -3619

Offset: 1

Christian G. Bower, May 16 2000

			1; -1,1; 1,-2,1; -1,4,-4,1; 0,-6,10,-5,1; ...

A055340 Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 8, 4, 1, 0, 1, 9, 19, 16, 5, 1, 0, 1, 12, 37, 46, 25, 6, 1, 0, 1, 16, 66, 118, 96, 40, 7, 1, 0, 1, 20, 110, 260, 300, 184, 56, 8, 1, 0, 1, 25, 172, 527, 811, 688, 318, 80, 9, 1, 0, 1, 30, 257, 985, 1951, 2178, 1408, 524, 105, 10, 1, 0

Offset: 1

Christian G. Bower, May 14 2000

			G.f. = x^(y + x*y + x^2*(y + y^2) + x^3*(y + 2*y^2 + y^3) + x^4*(y + 4*y^2 + 3*y^3 + y^4) + ...).
n\k 1  2  3  4  5  6  7  8
--:-- -- -- -- -- -- -- --
1:  1
2:  1  0
3:  1  1  0
4:  1  2  1  0
5:  1  4  3  1  0
6:  1  6  8  4  1  0
7:  1  9 19 16  5  1  0
8:  1 12 37 46 25  6  1  0

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Row sums give A032200.
Columns 2..8 are A002620(n-1), A055341, A055342, A055343, A055344, A055345, A055346.
Cf. A055347, A055348, A055349, A055356, A055363.

m = 13; A[, ] = 0;
Do[A[x_, y_] = x (y - Sum[EulerPhi[i]/i Log[1 - A[x^i, y^i]], {i, 1, m}]) + O[x]^m + O[y]^m // Normal, {m}];
Join[{1}, Append[CoefficientList[#/y, y], 0]& /@ Rest @ CoefficientList[ A[x, y]/x, x]] // Flatten (* Jean-François Alcover, Oct 02 2019 *)

{T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*y - x*sum(i=1, j, eulerphi(i)/i * log(1 - subst( subst( A + x * O(x^min(j, n\i)), x, x^i), y, y^i) ) )); polcoeff( polcoeff(A, n), k))}; /* Michael Somos, Aug 24 2015 */

G.f. satisfies A(x, y)=xy+x*CIK(A(x, y))-x. Shifts up under CIK transform.
G.f. satisfies A(x, y) = x*(y - Sum_{i>0} phi(i)/i * log(1 - A(x^i, y^i))). - Michael Somos, Aug 24 2015
Sum_k T(n, k) = A032200(n). - Michael Somos, Aug 24 2015

A055356 Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 18, 6, 0, 1, 26, 98, 96, 24, 0, 1, 57, 424, 874, 600, 120, 0, 1, 120, 1614, 6040, 8244, 4320, 720, 0, 1, 247, 5682, 35458, 83500, 83628, 35280, 5040, 0, 1, 502, 19022, 187288, 701164, 1169768, 915984, 322560, 40320, 0, 1

Offset: 1

Christian G. Bower, May 15 2000

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

Also related to the solution of the equation df/dt=f e^f (see the Maple code). - F. Chapoton, Jul 16 2004

			Triangle begins
  1;
  1,  0;
  1,  1,   0;
  1,  4,   2,   0;
  1, 11,  18,   6,   0;
  1, 26,  98,  96,  24,   0;
  1, 57, 424, 874, 600, 120, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - _N. J. A. Sloane_, Aug 21 2012
Index entries for sequences related to mobiles

Row sums give A029768 (p(n,1)).
Alternating row sums give A089963 (p(n+1,-1)).
Columns 2..8 are A000295, A055357, A055358, A055359, A055360, A055361, A055362.
Cf. A055340, A055349, A055363.

P[1]:=1;for n from 1 to 8 do P[n+1]:=simplify((1+n*x)*P[n]+x*diff(P[n],x)) end; # F. Chapoton, Jul 16 2004

P[1][_] = 1;
P[n_][x_] := P[n][x] = (1 + (n-1) x) P[n-1][x] + x P[n-1]'[x] // Expand;
row[1] = {1};
row[n_] := Append[CoefficientList[P[n-1][x], x], 0];
Array[row, 10] // Flatten (* Jean-François Alcover, Nov 17 2018, after F. Chapoton *)

A(n)={my(v=vector(n)); v[1]=y; for(n=2, #v, v[n]=v[n-1] + sum(k=1, n-2, binomial(n-2, k)*v[k]*v[n-k])); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018

Let p(n,x) be the polynomial with coefficients equal to the n-th row of the triangle in ascending powers of x, e.g., p(4,x) = 1+4*x+2*x^2; then p(n+1,x) = (1+(n-1)*x)*p(n,x) + x*p'(n,x). - Ben Whitmore, May 12 2021
Recurrence: T(n,k) = (n-2) * T(n-1,k-1) + k * T(n-1,k) for n >= 1, 1 <= k <= n with T(1,1) = 1 and T(n,k) = 0 for n < 1, k < 1 or k > n. - Georg Fischer, Oct 27 2021
Conjecture: row polynomials are R(n-2,0) for n > 1 where R(n,k) = R(n-1,k+1) + x*Sum_{i=0..n-1} Sum_{j=0..k} binomial(n-1, i)*R(n-i-1,j)*R(i,k-j) for n > 0, k >= 0 with R(0,k) = 1 for k >= 0. - Mikhail Kurkov, Apr 11 2025

A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 2, 0, 6, 3, 0, 24, 36, 8, 0, 120, 360, 220, 30, 0, 720, 3600, 4200, 1500, 144, 0, 5040, 37800, 71400, 47250, 11508, 840, 0, 40320, 423360, 1176000, 1234800, 545664, 98784, 5760, 0, 362880, 5080320, 19474560, 29635200, 20469456, 6618528, 940896, 45360, 0

Offset: 1

Christian G. Bower, May 15 2000

			Triangle begins:
     1;
     2,     0;
     6,     3,     0;
    24,    36,     8,     0;
   120,   360,   220,    30,     0;
   720,  3600,  4200,  1500,   144,   0;
  5040, 37800, 71400, 47250, 11508, 840, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Index entries for sequences related to mobiles

Row sums give A038037.
Columns 1..8 are A000142, A055303, A055350, A055351, A055352, A055353, A055354, A055355.
Cf. A055340, A055356, A055363.

T[rows_] := {{1}}~Join~((cc = CoefficientList[#, y]; Append[Rest[cc], 0] * Length[cc]!)& /@ (CoefficientList[InverseSeries[x/(y-Log[1-x + O[x]^rows] ), x], x][[3;;]]));
T[9] // Flatten (* Jean-François Alcover, Oct 31 2019 *)

A(n)={my(v=Vec(serlaplace(serreverse(x/(y - log(1-x + O(x^n))))))); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018

E.g.f. satisfies A(x, y) = x*y - x*log(1-A(x, y)). [Corrected by Sean A. Irvine, Mar 19 2022]

A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 23, 59, 148, 385, 1006, 2678, 7170, 19421, 52933, 145364, 401421, 1114713, 3109710, 8713076, 24506121, 69168705, 195849114, 556165311, 1583601840, 4520226558, 12931917204, 37075154703

Offset: 1

Christian G. Bower

Also the number of locally Lyndon plane trees with n nodes, where a plane tree is locally Lyndon if the sequence of branches directly under any given node is a Lyndon word. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 10 locally Lyndon plane trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  ((oo(o)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Wikipedia, Lyndon word
Index entries for sequences related to mobiles

Cf. A032200, A055363.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

T[n_, k_] := Module[{A}, A[, ] = 0; If[k < 1 || k > n, 0, For[j = 1, j <= n, j++, A[x_, y_] = x*y - x*Sum[MoebiusMu[i]/i * Log[1 -  A [x^i, y^i]] + O[x]^j // Normal , {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
a[n_] := a[n] = Sum[T[n, k], {k, 1, n}];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 28}] (* Jean-François Alcover, Jun 30 2017, using Michael Somos' code for A055363 *)
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
lynplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lynplane/@c],LyndonQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lynplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

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+CHK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CHK" (necklace, identity, unlabeled) transform.
From Petros Hadjicostas, Dec 03 2017: (Start)
a(n+1) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = c(1) = 1.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+1)*x^n = -Sum_{n>=1} (mu(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 + 51*x^5 + 147*x^6 + 414*x^7 + 1203*x^8 + ...
(End)

A055364 Number of asymmetric mobiles (circular rooted trees) with n nodes and 3 leaves.

Original entry on oeis.org

1, 4, 10, 22, 42, 73, 119, 184, 272, 389, 540, 731, 969, 1261, 1614, 2037, 2538, 3126, 3811, 4603, 5512, 6550, 7728, 9058, 10553, 12226, 14090, 16160, 18450, 20975, 23751, 26794, 30120, 33747, 37692, 41973, 46609, 51619, 57022, 62839, 69090, 75796, 82979

Offset: 5

Christian G. Bower, May 16 2000

nonn

			G.f. = x^5 + 4*x^6 + 10*x^7 + 22*x^8 + 42*x^9 + 73*x^10 + 119*x^11 + ...

Georg Fischer, Table of n, a(n) for n = 5..129
Index entries for sequences related to mobiles
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,0,1,2,-3,1).

Column 3 of A055363.

Cf. A055278.

a[ n_] := Quotient[ 3 n^4 - 28 n^3 + 102 n^2 - 180 n + 144, 144];

{a(n) = (3*n^4 - 28*n^3 + 102*n^2 - 180*n + 144) \ 144}; /* Michael Somos, Jun 29 2015 */

{a(n) = if( n<1, n = -n; polcoeff( (1 + x^2 + x^3) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)) + x * O(x^n), n), polcoeff( x^5 * (1 + x + x^3) / ((1 - x)^3 * (1 - x^2) * (1 - x^3)) + x * O(x^n), n))}; /* Michael Somos, Jun 29 2015 */

G.f.: x^5*(x^3+x+1)/((1-x^2)*(1-x^3)*(1-x)^3).

a(n) = A055278(4 - n) for all n in Z. - Michael Somos, Jun 29 2015

A055365 Number of asymmetric mobiles (circular rooted trees) with n nodes and 4 leaves.

Original entry on oeis.org

1, 5, 19, 53, 130, 280, 556, 1024, 1788, 2971, 4752, 7338, 11013, 16099, 23020, 32249, 44390, 60109, 80234, 105670, 137520, 176979, 225479, 284562, 356049, 441890, 544360, 665883, 809258, 977455, 1173871, 1402098, 1666212, 1970508, 2319825, 2719248, 3174469

Offset: 6

Christian G. Bower, May 16 2000

nonn

			G.f. = x^6 + 5*x^7 + 19*x^8 + 53*x^9 + 130*x^10 + 280*x^11 + 556*x^12 + ...

Georg Fischer, Table of n, a(n) for n = 6..129
Index entries for sequences related to mobiles

Column 4 of A055363.

Cf. A055279.

{a(n) = if( n<6, n = -n; polcoeff( (1 + x + 3*x^2 + 5*x^3 + 7*x^4 + 5*x^5 + 5*x^6 + 2*x^7 + x^8) / ((1 - x)^3 * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n), n = n-6; polcoeff( (1 + 2*x + 5*x^2 + 5*x^3 + 7*x^4 + 5*x^5 + 3*x^6 + x^7 + x^8) / ((1 - x)^3 * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n))}; /* Michael Somos, Nov 02 2014 */

G.f.: x^6*( -1-2*x-5*x^2-5*x^3-7*x^4-5*x^5-3*x^6-x^7-x^8 ) / ( (x^2+1)*(1+x+x^2)*(1+x)^3*(x-1)^7 ). - R. J. Mathar, Sep 18 2011
a(5-n) = A055279(n) for all n in Z. - Michael Somos, Nov 02 2014
0 = -30 + a(n) - 2*a(n+1) - a(n+2) + 3*a(n+3) + a(n+5) - 2*a(n+6) - 2*a(n+7) + a(n+8) + 3*a(n+10) - a(n+11) - 2*a(n+12) + a(n+13) for all n in Z. - Michael Somos, Nov 02 2014
a(n) ~ n^6 / 1152 as n -> infinity. - Michael Somos, Nov 02 2014

A055366 Number of asymmetric mobiles (circular rooted trees) with n nodes and 5 leaves.

Original entry on oeis.org

1, 7, 31, 109, 321, 833, 1951, 4212, 8498, 16205, 29449, 51349, 86363, 140731, 222986, 344607, 520758, 771208, 1121360, 1603501, 2258188, 3135900, 4298854, 5823139, 7801042, 10343750, 13584274, 17680810, 22820358, 29222841

Offset: 7

Christian G. Bower, May 16 2000

nonn

Georg Fischer, Table of n, a(n) for n = 7..129
Index entries for sequences related to mobiles

Column 5 of A055363.

A055367 Number of asymmetric mobiles (circular rooted trees) with n nodes and 6 leaves.

Original entry on oeis.org

1, 8, 45, 190, 672, 2045, 5580, 13868, 31968, 69069, 141284, 275484, 515296, 928952, 1620895, 2746538, 4533060, 7305216, 11520492, 17811767, 27043799, 40380695, 59373007, 86060337, 123101225, 173924181, 242915666, 335636501

Offset: 8

Christian G. Bower, May 16 2000

nonn

Georg Fischer, Table of n, a(n) for n = 8..129
Index entries for sequences related to mobiles

Column 6 of A055363.

A055368 Number of asymmetric mobiles (circular rooted trees) with n nodes and 7 leaves.

Original entry on oeis.org

1, 10, 63, 310, 1260, 4423, 13772, 38876, 101036, 244770, 558083, 1206992, 2491908, 4937234, 9429273, 17423919, 31251810, 54558739, 92928819, 154753337, 252422630, 403940910, 635084757, 982257860, 1496223663

Offset: 9

Christian G. Bower, May 16 2000

nonn

Georg Fischer, Table of n, a(n) for n = 9..129
Index entries for sequences related to mobiles

Column 7 of A055363.

cp's OEIS Frontend

A055370 Matrix inverse of triangle A055363(n+2,k).

Views

Author

Keywords

Examples

A055340 Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A055356 Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A055364 Number of asymmetric mobiles (circular rooted trees) with n nodes and 3 leaves.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A055365 Number of asymmetric mobiles (circular rooted trees) with n nodes and 4 leaves.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A055366 Number of asymmetric mobiles (circular rooted trees) with n nodes and 5 leaves.

Views

Author

Keywords

Links