A133386 -id:A133386 - cp's OEIS frontend

A133399 Triangle T(n,k)=number of forests of labeled rooted trees with n nodes, containing exactly k trees of height one, all others having height zero (n>=0, 0<=k<=floor(n/2)).

1, 1, 1, 2, 1, 9, 1, 28, 12, 1, 75, 120, 1, 186, 750, 120, 1, 441, 3780, 2100, 1, 1016, 16856, 21840, 1680, 1, 2295, 69552, 176400, 45360, 1, 5110, 272250, 1224720, 705600, 30240, 1, 11253, 1026300, 7692300, 8316000, 1164240, 1, 24564, 3762132, 45018600

Offset: 0

Alois P. Heinz, Nov 24 2007

			Triangle begins:
  1;
  1;
  1,     2;
  1,     9;
  1,    28,     12;
  1,    75,    120;
  1,   186,    750,     120;
  1,   441,   3780,    2100;
  1,  1016,  16856,   21840,   1680;
  1,  2295,  69552,  176400,  45360;
  1,  5110, 272250, 1224720, 705600, 30240;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
A. P. Heinz, Finding Two-Tree-Factor Elements of Tableau-Defined Monoids in Time O(n^3), Ed. S. G. Akl, F. Fiala, W. W. Koczkodaj: Advances in Computing and Information, ICCI90 Niagara Falls, LNCS 468, Springer-Verlag (1990), pp. 120-128.

Columns k=1,2 give: A058877, A133386.
Row sums give: A000248.
T(2n,n) = A001813(n), T(2n+1,n) = A002691(n).
Reading the table by diagonals gives triangle A198204. - Peter Bala, Jul 31 2012
Cf. A235596.

/* As triangle */ [[Binomial(n,k)*Factorial(k)*StirlingSecond(n-k+1,k+1): k in [0..Floor(n/2)]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 06 2019

T:= (n,k)-> binomial(n,k)*k!*Stirling2(n-k+1,k+1): for n from 0 to 10 do lprint(seq(T(n, k), k=0..floor(n/2))) od;

nn=12;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[ Series[Exp[y x (Exp[x]-1)] Exp[x],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Feb 09 2013 *)
t[n_, k_] := Binomial[n, k]*k!*StirlingS2[n-k+1, k+1]; Table[t[n, k], {n, 0, 12}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

T(n,k) = C(n,k) * k! * stirling2(n-k+1,k+1).
E.g.f.: exp(y*x*(exp(x)-1))*exp(x). - Geoffrey Critzer, Feb 09 2013
Sum_{k=1..floor(n/2)} T(n,k) = A235596(n+1). - Alois P. Heinz, Jun 21 2019

A198204 Series reversion of (1 - tx)log(1 + x) with respect to x.

Original entry on oeis.org

1, 1, 2, 1, 9, 12, 1, 28, 120, 120, 1, 75, 750, 2100, 1680, 1, 186, 3780, 21840, 45360, 30240, 1, 441, 16856, 176400, 705600, 1164240, 665280, 1, 1016, 69552, 1224720, 8316000, 25280640, 34594560, 17297280, 1, 2295, 272250, 7692300, 82577880, 408648240, 998917920, 1167566400, 518918400

Offset: 1

Peter Bala, Jul 31 2012

This triangle is A133399 read by diagonals.

			Triangle begins
.n\k.|..0....1.....2......3......4......5
= = = = = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....2
..3..|..1....9....12
..4..|..1...28...120....120
..5..|..1...75...750...2100...1680
..6..|..1..186..3780..21840..45360..30240
...

Cf. A001813, A002691, A052892, A058877, A133386, A133399, A141618.

Flatten[CoefficientList[CoefficientList[InverseSeries[Series[Log[1 + x]*(1 - t*x),{x,0,9}]], x]*Table[n!, {n,0,9}], t]] (* Peter Luschny, Oct 25 2015 *)

T(n,k) = k!*binomial(n + k - 1,k)*Stirling2(n,k + 1) (n >= 1, k >=0).
E.g.f.: A(x,t) = series reversion of (1 - t*x)*log(1 + x) w.r.t. x = x + (1 + 2*t)*x^2/2! + (1 + 9*t + 12*t^2)*x^3/3! + ....
Main diagonal A001813, first subdiagonal A002691.
Column 1 A058877, column 2 A133386. Row sums A052892.
1 - t*A(x,t) = x/series reversion of x*(1 - t(exp(x) - 1)) with respect to x. Cf. A141618. - Peter Bala, Oct 22 2015

cp's OEIS Frontend

A133399 Triangle T(n,k)=number of forests of labeled rooted trees with n nodes, containing exactly k trees of height one, all others having height zero (n>=0, 0<=k<=floor(n/2)).

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A198204 Series reversion of (1 - tx)log(1 + x) with respect to x.

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cp's OEIS Frontend

A133399 Triangle T(n,k)=number of forests of labeled rooted trees with n nodes, containing exactly k trees of height one, all others having height zero (n>=0, 0<=k<=floor(n/2)).

Views

Author

Keywords

Examples

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Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A198204 Series reversion of (1 - t*x)*log(1 + x) with respect to x.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A198204 Series reversion of (1 - tx)log(1 + x) with respect to x.