A055349 -id:A055349 - cp's OEIS frontend

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

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

A055363 Triangle of asymmetric mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 6, 10, 5, 1, 0, 0, 1, 9, 22, 19, 7, 1, 0, 0, 1, 12, 42, 53, 31, 8, 1, 0, 0, 1, 16, 73, 130, 109, 45, 10, 1, 0, 0, 1, 20, 119, 280, 321, 190, 63, 11, 1, 0, 0, 1, 25, 184, 556, 833, 672, 310, 83, 13, 1, 0, 0, 1, 30, 272

Offset: 1

Christian G. Bower, May 15 2000

			G.f. = x*(y + x*y + x^2*y + x^3*(y + y^2) + x^4*(y + 2*y^2 + y^3) + x^5*(y + 4*y^2 + 4*y^3 + y^4) + ...).
n\k 1  2  3  4  5  6  7  8
--:-- -- -- -- -- -- -- --
1:  1
2:  1  0
3:  1  0  0
4:  1  1  0  0
5:  1  2  1  0  0
6:  1  4  4  1  0  0
7:  1  6 10  5  1  0  0
8:  1  9 22 19  7  1  0  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 A032171.
Columns 2..8: A002620(n-2), A055364, A055365, A055366, A055367, A055368, A055369.
Cf. A055340, A055349, A055356, A055370, A055371.

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]]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 30 2017, after Michael Somos *)

{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, moebius(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) = x*(y - Sum_{i>0} moebius(i)/i * log(1 - A(x^i, y^i))). - Michael Somos, Aug 19 2015

Sum_k T(n, k) = A032171(n). - Michael Somos, Aug 24 2015

A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.

Original entry on oeis.org

1, 2, 9, 68, 730, 10164, 173838, 3524688, 82627200, 2198295360, 65431163160, 2154106470240, 77714083773456, 3048821300491680, 129221979665461200, 5884296038166954240, 286492923374605966080, 14851359950834255500800

Offset: 1

Christian G. Bower, Sep 15 1998

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.83).

Alois P. Heinz, Table of n, a(n) for n = 1..140
C. G. Bower, Transforms (2)
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 454
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Index entries for sequences related to mobiles

Cf. A029768, A032200, A055349.

logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: b:= logtr(-a): a:= n-> `if`(n<=1,1, -n*b(n-1)): seq(a(n), n=1..25); # Alois P. Heinz, Sep 14 2008

a[n_] = Sum[Binomial[n, j]*Abs[StirlingS1[n-1, j]]*j!, {j, 0, n}]; Array[a, 18]
(* Jean-François Alcover, Jun 22 2011, after Vladimir Kruchinin *)

Vec(serlaplace(serreverse(x/(1 - log(1-x + O(x^20)))))) \\ Andrew Howroyd, Sep 19 2018

Divides by n and shifts left under "CIJ" (necklace, indistinct, labeled) transform.
E.g.f. A(x) satisfies A(x) = x-x*log(1-A(x)). [Corrected by Andrey Zabolotskiy, Sep 16 2022]
a(n) = Sum_{j=0..n} binomial(n,j)*abs(Stirling1(n-1,j))*j!, n > 0. - Vladimir Kruchinin, Feb 03 2011
a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^(n-1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013
E.g.f.: series reversion of x/(1 - log(1-x)). - Andrew Howroyd, Sep 19 2018

A055350 Number of labeled mobiles (circular rooted trees) with n nodes and 3 leaves.

Original entry on oeis.org

8, 220, 4200, 71400, 1176000, 19474560, 330220800, 5787936000, 105380352000, 1997835840000, 39477236198400, 813155511168000, 17453093898240000, 390070546145280000, 9070029416448000000, 219204470936715264000

Offset: 4

Christian G. Bower, May 15 2000

nonn

Georg Fischer, Table of n, a(n) for n = 4..250
Index entries for sequences related to mobiles

Column 3 of A055349.

m = 20;
L[x_] = Log[1-x] + O[x]^m // Normal;
A[_] = 0;
Do[A[x_] = x y - x L[A[x]] + O[x]^k // Normal // Expand, {k, m}];
Drop[(SeriesCoefficient[#, {y, 0, 3}]& /@ CoefficientList[A[x], x]) * Range[0, m-1]!, 4] (* Jean-François Alcover, Nov 01 2019 *)

A055351 Number of labeled mobiles (circular rooted trees) with n nodes and 4 leaves.

Original entry on oeis.org

30, 1500, 47250, 1234800, 29635200, 685843200, 15717240000, 362244960000, 8476532064000, 202580554176000, 4963223577312000, 124948285862400000, 3236755595673600000, 86346680044584960000, 2372991796225290240000

Offset: 5

Christian G. Bower, May 15 2000

nonn

Georg Fischer, Table of n, a(n) for n = 5..250
Index entries for sequences related to mobiles

Column 4 of A055349.

m = 20; L[x_] = Log[1 - x] + O[x]^m // Normal; A[_] = 0;
Do[A[x_] = x y - x  L[A[x]] + O[x]^k // Normal // Expand, {k, m}];
Drop[(SeriesCoefficient[#, {y, 0, 4}] & /@ CoefficientList[A[x], x]) * Range[0, m - 1]!, 5] (* Jean-François Alcover, Nov 01 2019 *)

A055352 Number of labeled mobiles (circular rooted trees) with n nodes and 5 leaves.

Original entry on oeis.org

144, 11508, 545664, 20469456, 678857760, 21047130720, 629779328640, 18547337128320, 544474197146880, 16067754570067200, 479436489384652800, 14522782898521497600, 447847214484186316800, 14087092606079728435200

Offset: 6

Christian G. Bower, May 15 2000

nonn

Georg Fischer, Table of n, a(n) for n = 6..250
Index entries for sequences related to mobiles

Column 5 of A055349.

m = 20; L[x_] = Log[1 - x] + O[x]^m // Normal; A[_] = 0;
Do[A[x_] = x y - x L[A[x]] + O[x]^k // Normal // Expand, {k, m}];
Drop[(SeriesCoefficient[#, {y, 0, 5}] & /@ CoefficientList[A[x], x]) * Range[0, m - 1]!, 6] (* Jean-François Alcover, Nov 01 2019 *)

A055353 Number of labeled mobiles (circular rooted trees) with n nodes and 6 leaves.

Original entry on oeis.org

840, 98784, 6618528, 339111360, 14931378000, 600119150400, 22811289621120, 838683494288640, 30275663821603200, 1084273458428160000, 38808699592126464000, 1395650171787308236800, 50626461643690886553600

Offset: 7

Christian G. Bower, May 15 2000

nonn

Georg Fischer, Table of n, a(n) for n = 7..250
Index entries for sequences related to mobiles

Column 6 of A055349.

m = 20; L[x_] = Log[1 - x] + O[x]^m // Normal; A[_] = 0;
Do[A[x_] = x y - x L[A[x]] + O[x]^k // Normal // Expand, {k, m}];
Drop[(SeriesCoefficient[#, {y, 0, 6}] & /@ CoefficientList[A[x], x]) * Range[0, m - 1]!, 7] (* Jean-François Alcover, Nov 01 2019 *)

A055354 Number of labeled mobiles (circular rooted trees) with n nodes and 7 leaves.

Original entry on oeis.org

5760, 940896, 85049280, 5731545600, 324745027200, 16481262283200, 778208622871680, 35036455255401600, 1529404526377728000, 65498410076875776000, 2775672164602681344000, 117137057625636739891200

Offset: 8

Christian G. Bower, May 15 2000

nonn

Georg Fischer, Table of n, a(n) for n = 8..250
Index entries for sequences related to mobiles

Column 7 of A055349.

m = 20; L[x_] = Log[1 - x] + O[x]^m // Normal; A[_] = 0;
Do[A[x_] = x y - x  L[A[x]] + O[x]^k // Normal // Expand, {k, m}];
Drop[(SeriesCoefficient[#, {y, 0, 7}] & /@ CoefficientList[A[x], x]) * Range[0, m - 1]!, 8] (* Jean-François Alcover, Nov 01 2019 *)

A055355 Number of labeled mobiles (circular rooted trees) with n nodes and 8 leaves.

Original entry on oeis.org

45360, 9862560, 1160973000, 99904860000, 7103580541200, 445556635524000, 25652184715357200, 1391455702596960000, 72404516222438400000, 3661224054771419136000, 181631020686490222848000

Offset: 9

Christian G. Bower, May 15 2000

nonn

Georg Fischer, Table of n, a(n) for n = 9..250
Index entries for sequences related to mobiles

Column 8 of A055349.

m = 20; L[x_] = Log[1 - x] + O[x]^m // Normal; A[_] = 0;
Do[A[x_] = x y - x  L[A[x]] + O[x]^k // Normal // Expand, {k, m}];
Drop[(SeriesCoefficient[#, {y, 0, 8}] & /@ CoefficientList[A[x], x]) * Range[0, m - 1]!, 9] (* Jean-François Alcover, Nov 01 2019 *)

cp's OEIS Frontend

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

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

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A055363 Triangle of asymmetric mobiles (circular rooted trees) with n nodes and k leaves.

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A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.

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A055350 Number of labeled mobiles (circular rooted trees) with n nodes and 3 leaves.

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A055351 Number of labeled mobiles (circular rooted trees) with n nodes and 4 leaves.

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A055352 Number of labeled mobiles (circular rooted trees) with n nodes and 5 leaves.

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A055353 Number of labeled mobiles (circular rooted trees) with n nodes and 6 leaves.

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