A055303 -id:A055303 - cp's OEIS frontend

A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.

1, 2, 0, 6, 3, 0, 24, 36, 4, 0, 120, 360, 140, 5, 0, 720, 3600, 3000, 450, 6, 0, 5040, 37800, 54600, 18900, 1302, 7, 0, 40320, 423360, 940800, 588000, 101136, 3528, 8, 0, 362880, 5080320, 16087680, 15876000, 5143824, 486864, 9144, 9, 0, 3628800

Offset: 1

Christian G. Bower, May 11 2000

Beginning with the second row, dividing each row by n gives the mirror of row n-1 of A141618. Under the exponential transform, the mirror of A141618 is generated, relating the number of connected graphs here to the number of disconnected graphs associated with A141618 (cf. A127671 and A036040). - Tom Copeland, Oct 25 2014

			Triangle begins
     1,
     2,     0;
     6,     3,     0;
    24,    36,     4,     0;
   120,   360,   140,     5,    0;
   720,  3600,  3000,   450,    6, 0;
  5040, 37800, 54600, 18900, 1302, 7, 0;

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 313.

Alois P. Heinz, Rows n = 1..141, flattened
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Row sums give A000169. Columns 1 through 12: A000142, A055303-A055313. Cf. A055314.

Cf. A248120 for a natural refinement.

T:= (n, k)-> (n!/k!)*Stirling2(n-1, n-k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Nov 13 2013

Table[Table[n!/k! StirlingS2[n-1,n-k], {k,1,n}], {n,0,10}]//Grid  (* Geoffrey Critzer, Dec 01 2012 *)

A055302(n,k)=n!/k!*stirling(n-1, n-k,2);
for(n=1,10,for(k=1,n,print1(A055302(n,k),", "));print());
\\ Joerg Arndt, Oct 27 2014

E.g.f. (relative to x) satisfies: A(x,y) = xy + x*exp(A(x,y)) - x. Divides by n and shifts up under exponential transform.
T(n,k) = (n!/k!)*Stirling2(n-1, n-k). - Vladeta Jovovic, Jan 28 2004
T(n,k) = A055314(n,k)*(n-k) + A055314(n,k+1)*(k+1). The first term is the number of such trees with root degree > 1 while the second term is the number of such trees with root degree = 1. This simplifies to the above formula by Vladeta Jovovic. - Geoffrey Critzer, Dec 01 2012
E.g.f.: G(x,t) = log[1 + t * N(x*t,1/t)], where N(x,t) is the e.g.f. of A141618. Also, G(x*t,1/t)= log[1 + N(x,t)/t] is the comp. inverse in x of x / [1 + t * (e^x - 1)]. - Tom Copeland, Oct 26 2014

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]

A202363 Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.

Original entry on oeis.org

1, 6, 3, 36, 24, 12, 240, 180, 120, 60, 1800, 1440, 1080, 720, 360, 15120, 12600, 10080, 7560, 5040, 2520, 141120, 120960, 100800, 80640, 60480, 40320, 20160, 1451520, 1270080, 1088640, 907200, 725760, 544320, 362880, 181440, 16329600, 14515200, 12700800, 10886400, 9072000, 7257600, 5443200, 3628800, 1814400

Offset: 2

Geoffrey Critzer, Jan 09 2013

Row sums = A001809.

Column for k = 0 is A001286.

			T(3,1) = 3 because from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have respectively 3 inversion pairs (1,2), (2,3) and (1,3) which are all separated by 1 element.
Triangle T(n,k) begins:
       1;
       6,      3;
      36,     24,     12;
     240,    180,    120,    60;
    1800,   1440,   1080,   720,   360;
   15120,  12600,  10080,  7560,  5040,  2520;
  141120, 120960, 100800, 80640, 60480, 40320, 20160;
  ...

Alois P. Heinz, Rows n = 2..142, flattened

Cf. A001286, A001809, A055303.

nn=10;Range[0,nn]!CoefficientList[Series[x^2/2/(1-x)^2/(1-y x),{x,0,nn}],{x,y}]//Grid

E.g.f.: x^2/2 * (1/(1-x)^2)* (1/(1-y*x)).

A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 12, 6, 0, 4, 36, 72, 24, 0, 5, 80, 360, 480, 120, 0, 6, 150, 1200, 3600, 3600, 720, 0, 7, 252, 3150, 16800, 37800, 30240, 5040, 0, 8, 392, 7056, 58800, 235200, 423360, 282240, 40320, 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880

Offset: 0

Peter Luschny, Mar 28 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,   2;
  [3] 0, 3,  12,     6;
  [4] 0, 4,  36,    72,     24;
  [5] 0, 5,  80,   360,    480,     120;
  [6] 0, 6, 150,  1200,   3600,    3600,     720;
  [7] 0, 7, 252,  3150,  16800,   37800,   30240,    5040;
  [8] 0, 8, 392,  7056,  58800,  235200,  423360,  282240,   40320;
  [9] 0, 9, 576, 14112, 169344, 1058400, 3386880, 5080320, 2903040, 362880;

Cf. A052852 (row sums), A317365 (alternating row sums), A000142 (main diagonal), A187535 (central column), A062119, A055303, A011379.

Cf. A144084, A021010.

A361893 := (n, k) -> n!*binomial(n - 1, k - 1)/(n - k)!:
seq(seq(A361893(n,k), k = 0..n), n = 0..9);
# Using the egf.:
egf := 1 + (x*y/(1 - x*y))*exp(y/(1 - x*y)): ser := series(egf, y, 10):
poly := n -> convert(n!*expand(coeff(ser, y, n)), polynom):
row := n -> seq(coeff(poly(n), x, k), k = 0..n): seq(print(row(n)), n = 0..6);

T(n, k) = k! * binomial(n, k) * binomial(n - 1, k - 1).
T(n + 1, k + 1) / (n + 1) = A144084(n, k) = (-1)^(n - k)*A021010(n, k).
T(n, k) = [x^k] n! * ([y^n](1 + (x*y / (1 - x*y)) * exp(y / (1 - x*y)))).

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A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.

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

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A202363 Triangular array read by rows: T(n,k) is the number of inversion pairs ( p(i) < p(j) with i>j ) that are separated by exactly k elements in all n-permutations (where the permutation is represented in one line notation); n>=2, 0<=k<=n-2.

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A361893 Triangle read by rows. T(n, k) = n! * binomial(n - 1, k - 1) / (n - k)!.

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