A002866 -id:A002866 - cp's OEIS frontend

A178886 Divide, cell by cell, sequence A178802 by sequence A178888.

1, 1, 1, 2, 2, 1, 6, 4, 2, 3, 1, 24, 12, 12, 6, 6, 4, 1, 120, 48, 48, 24, 18, 36, 6, 8, 12, 5, 1, 720, 240, 240, 240, 72, 144, 72, 72, 24, 72, 24, 10, 20, 6, 1, 5040, 1440, 1440, 1440, 720, 360

Offset: 1

Alford Arnold, Jun 21 2010

Sequences A178802, A178888 and A178886 have A000041(n) entries per row.

			A178802 begins 1 2 2 6 12 6 24 48 24 72 24 ...
A178888 begins 1 2 2 3 6 6 4 12 12 24 24 ...
therefore
A178886 begins 1 1 1 2 2 1 6 4 2 3 1 ...

Cf. A000041, A000522 (row sums), A002866, A178802, A178887, A178888.

T(n,k) = A178802(n,k) / A178888(n,k)

A194649 Triangle of coefficients of a sequence of polynomials related to the enumeration of linear labeled rooted trees.

Original entry on oeis.org

1, 1, 3, 4, 13, 36, 24, 75, 316, 432, 192, 541, 3060, 6360, 5760, 1920, 4683, 33244, 92880, 127680, 86400, 23040, 47293, 403956, 1418424, 2620800, 2688000, 1451520, 322560, 545835, 5449756, 23051952, 53548992, 73785600, 60318720, 27095040, 5160960, 7087261, 80985780, 400813080, 1122145920, 1943867520, 2133734400

Offset: 0

Peter Bala, Sep 01 2011

Define the sequence of polynomials {P(n,x)}n>=0 recursively by setting P(0,x) = 1, P(1,x) = 1 and P(n+1,x) = d/dx((1+x)*(1+2*x)*P(n,x)) for n >= 1. The first few values are P(2,x) = 3 + 4*x, P(3,x) = 13 + 36*x + 24*x^2 and P(4,x) = 75 + 316*x + 432*x^2 + 192*x^3.
This triangle shows the coefficients of the P(n,x) in ascending powers of x. The values of P(n,x) at an integer or half-integer value of x enumerate linear labeled rooted trees: in particular we have P(n,0) = A000670(n), P(n,1/2) = A050351(n), P(n,1) = A050352(n) and P(n,3/2) = A050353(n).
More generally, for m >= 2, P(n,m/2-1), n = 0,1,2,... counts m level linear labeled rooted trees (see the e.g.f. below and the comment of Benoit Cloitre in A050351).

			Triangle begins
n\k|......0.......1........2........3........4........5.......6
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
.0.|......1
.1.|......1
.2.|......3.......4
.3.|.....13......36.......24
.4.|.....75.....316......432......192
.5.|....541....3060.....6360.....5760.....1920
.6.|...4683...33244....92880...127680....86400....23040
.7.|..47293..403956..1418424..2620800..2688000..1451520..322560
..

L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.

Cf. A000670, A002866 (main diagonal), A050351, A050352, A050353, A083411 (1/4*column 1).

T[0, 0] = T[1, 0] = 1; T[n_, k_] /; 0 <= k <= n-1 := T[n, k] = (k+1)*(2* T[n-1, k-1] + 3*T[n-1, k] + T[n-1, k+1]); T[, ] = 0;
{1}~Join~Table[T[n, k], {n, 1, 9}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 13 2019 *)

T(n,k) = 2^k*Sum_{i = k+1..n} Stirling2(n,i)*i!*binomial(i-1,k).
Recurrence: T(n+1,k) = (k+1)*(2*T(n,k-1)+3*T(n,k)+T(n,k+1)).
E.g.f.: G(x,t) := 1 + (1-exp(t))/((2*x+1)*exp(t)-2*x-2) = Sum_{n>=0} P(n,x)*t^n/n! = 1 + t + (3 + 4*x)*t^2/2! + (13 + 36*x + 24*x^2)*t^3/3! + ....
Column k generating function: 2^k*((exp(x)-1)/(2-exp(x)))^(k+1) (apart from initial term 1 when k = 0).
The generating function G(x,t) satisfies the partial differential equation d/dx((1+x)*(1+2*x)*G(x,t)) - d/dt(G(x,t)) = 2*(2x+1). Hence the row polynomials P(n,x) satisfy the defining recurrence P(n+1,x) = d/dx((1+x)*(2+x)*P(n,x)), with P(0,x) = P(1,x) = 1.
Reflection property: P(n,x) = (-1)^n*P(n,-x-3/2).
The polynomial P(n,x) has all real zeros, lying in the interval [-1,-1/2] (apply [Liu et al, Theorem 1.1, Corollary 1.2] with f(x) = P(n,x-1/2) and g(x) = P'(n,x-1/2) and use the reflection property).
Row sums are A050352; Column 0: A000670; Column 1: 4*A083411; Main diagonal: A002866.

A208057 Triangle by rows, generated from the odd integers and related to A000165.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 24, 18, 5, 1, 192, 144, 40, 7, 1, 1920, 1440, 400, 70, 9, 1, 23040, 17280, 4800, 840, 108, 11, 1, 322560, 241920, 67200, 11760, 1512, 154, 13, 1, 5160960, 3870720, 1075200, 188160, 24192, 2464, 208, 15, 1

Offset: 0

Gary W. Adamson, Feb 22 2012

Row sums = A000165, the double factorial numbers: (1, 2, 8, 48, 384,...).

Left border = A002866 and the eigensequence of the odd integers prefaced with a 1.

			First few rows of the triangle:
  1;
  1, 1;
  4, 3, 1;
  24, 18, 5, 1;
  192, 144, 40, 7, 1;
  1920, 1440, 400, 70, 9, 1;
  23040, 17280, 4800, 840, 108, 11, 1;
  322560, 241920, 67200, 11760, 1512, 154, 13, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000165, A002866, A158405.

T:= proc(n) option remember; local M;
      M:= (Matrix(n+1, (i, j)-> `if`(i=j, 1, `if`(i>j, -2*j+1, 0)))^(-1));
      seq(M[n+1, k], k=1..n+1)
    end:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 27 2012

T[n_] := T[n] = Module[{M}, M = Table[If[i == j, 1, If[i>j, -2*j+1, 0]], {i, 1, n+1 }, {j, 1, n+1}] // Inverse; M[[n+1]]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

Eigentriangle of triangle A158405 (odd integers in every row: (1, 3, 5,...)); the inverse of:
   1;
  -1,  1;
  -1, -3,  1;
  -1, -3, -5,  1;
  -1, -3, -5, -7,  1;
  ...

Typo in term 17 corrected by Alois P. Heinz, Dec 06 2012

A302606 a(n) = n! * [x^n] exp(nx)arcsinh(x).

Original entry on oeis.org

0, 1, 4, 26, 240, 2884, 42660, 748544, 15185856, 349574544, 9000902500, 256293989984, 7996078704240, 271246034903232, 9939835626507332, 391303051339622400, 16469438021801262848, 737992773619777599744, 35077254665501330210628, 1762671472887447792620032

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

N. J. A. Sloane, Transforms

Cf. A001818, A002866, A291483, A302583, A302584, A302585, A302586, A302587, A302605, A302608, A302609.

Table[n! SeriesCoefficient[Exp[n x] ArcSinh[x], {x, 0, n}], {n, 0, 19}]

a(n) ~ arcsinh(1) * n^n = log(1 + sqrt(2)) * n^n. - Vaclav Kotesovec, Jun 09 2019

a(n) = Sum_{k=1..n, k odd} (-1)^((k-1)/2)*binomial(n,k)*(k-2)!!^2*n^(n-k). - Fabian Pereyra, Oct 05 2024

A308939 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} (2k - 1)!!x^k/k!).

Original entry on oeis.org

1, 1, 5, 39, 411, 5445, 86805, 1616895, 34448715, 826093485, 22017673125, 645633501975, 20655688959675, 715958472554325, 26726481024167925, 1068988088284491375, 45608095005687088875, 2067503007329827192125, 99238033465208117605125, 5027986481205385725402375

Offset: 0

Ilya Gutkovskiy, Jul 01 2019

nonn

Cf. A001147, A002866, A295553.

a := n -> local j, k; 2^n*add(add((-1)^(k-j)*binomial(k, j)*pochhammer(j/2, n), j = 0..k), k = 0..n): seq(a(n), n = 0..19);  # Peter Luschny, Mar 08 2024

nmax = 19; CoefficientList[Series[1/(2 - 1/Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (2 k - 1)!! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

E.g.f.: 1/(2 - 1/sqrt(1 - 2*x)).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2*k - 1)!! * a(n-k).
a(n) ~ n! * 8^n / 3^(n+1). - Vaclav Kotesovec, Jul 01 2019
D-finite with recurrence: +3*a(n) +(-14*n+9)*a(n-1) +8*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 27 2020
a(n) = 2^n*Sum_{k=0..n} Sum_{j=0..k} (-1)^(k-j)*binomial(k, j)*Pochhammer(j/2, n). - Peter Luschny, Mar 08 2024

A001217 Sorted list of orders of Weyl groups of types A_n, B_n, D_n, E_n, F_4, G_2.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 48, 120, 192, 384, 720, 1152, 1920, 3840, 5040, 23040, 40320, 46080, 51840, 322560, 362880, 645120, 2903040, 3628800, 5160960, 10321920, 39916800, 92897280, 185794560, 479001600, 696729600, 1857945600, 3715891200

Offset: 1

N. J. A. Sloane

N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6.

O(A_n) = (n+1)! = A000142(n+1), O(B_n) = O(C_n) = 2^n*n! = A000165(n), O(D_n) = 2^(n-1)*n! = A002866(n), O(E_n) = A003134(n), O(F_4) = 1152, O(G_2) = 12. Hence, {A001217} = {A000142} union {A000165} union {A002866} union {A003134} union {12, 1152}. - Sean A. Irvine, Jul 11 2019

A028371 a(n) = (2^(n-1)-1)!*2^((2^(n-1)-1)-1).

Original entry on oeis.org

1, 24, 322560, 21424936845312000, 8829205774994708066835865418197893120000000

Offset: 2

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 2..9
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.

Cf. A002866.

[(Factorial(2^(n-1)-1))*2^((2^(n-1)-1)-1): n in [2..8]]; // Vincenzo Librandi, Jun 03 2011

a(n) = A002866(2^(n-1)-1).

A052564 Expansion of e.g.f. x(1-x)/(1-2x).

Original entry on oeis.org

0, 1, 2, 12, 96, 960, 11520, 161280, 2580480, 46448640, 928972800, 20437401600, 490497638400, 12752938598400, 357082280755200, 10712468422656000, 342798989524992000, 11655165643849728000, 419585963178590208000

Offset: 0

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

Partition the set {1,2,...,n} into an odd number of subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. - Geoffrey Critzer, Mar 05 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 506

Essentially the same as A014297.

[n le 1 select n else 2^(n-2)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 05 2019

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

a = x/(1 - x); CoefficientList[Series[a/(1 - a^2), {x, 0, 20}], x]* Table[n!, {n, 0, 20}] (* Geoffrey Critzer, Mar 05 2010 *)
Part[#,Range[1, Length[#], 1]]&@(Array[#!&, Length[#], 0]*#)&@CoefficientList[Series[x*(1-x)/(1-2x), {x, 0, 20}], x]// ExpandAll (* Vincenzo Librandi, Jan 04 2013 - after Olivier Gérard in A213068 *)
With[{nn=20},CoefficientList[Series[x (1-x)/(1-2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2025 *)

{a(n) = if(n<=1, n, 2^(n-2)*n!)}; \\ G. C. Greubel, May 05 2019

my(x='x+O('x^20)); concat([0], Vec(serlaplace(x*(1-x)/(1-2*x)))) \\ Felix Fröhlich, May 05 2019

[0,1]+[2^(n-2)*factorial(n) for n in (2..20)] # G. C. Greubel, May 05 2019

E.g.f.: x*(1-x)/(1-2*x).
a(n) = 2*n*a(n-1), with a(0)=0, a(1)=1, a(2)=2.
a(n) = 2^(n-2) * n! for n>1.
a(n) = A002866(n) - A014297(n-2) for n>1. - Geoffrey Critzer, Mar 05 2010

A072503 Number of ways to lace a shoe with n eyelet pairs such that there is no direct "horizontal" connection between any adjacent eyelet pair.

Original entry on oeis.org

3, 45, 1824, 109560, 9520560, 1145057760, 181091917440

Offset: 3

Hugo Pfoertner, Jan 27 2003

The lacing must not have any "straight connections" between adjacent eyelet pairs (e.g. 2<->2*n-1, 3<->2*n-2, 4<->2*n-3,....). There are no symmetric solutions.
From Sean A. Irvine, Oct 06 2024: (Start)
The lacing must begin and end with the top eyelet pair (1 and 2n in Pfoertner's numbering).
Every eyelet must be used.
Under versus over crossings are not considered distinct.
Three consecutive eyelets on the same side are not permitted.
Because there are no symmetric solutions, mirrored solutions can be handled by simply dividing the total solutions by 2. (End)

			The 6 non-straight lacings for n=3 are: 124536, 135426, 142356, 145326, 153246, 154236. Not counting mirror images we get a(3)=3.

Sean A. Irvine, Java program (github)
Hugo Pfoertner, FORTRAN program to count non-straight shoe lacings and results for N=3,4
Index entries for sequences related to shoe lacings

Cf. A078602, A078698, A078702, A002866.

Fortran
```
See Links.
```

a(9) from Sean A. Irvine, Oct 07 2024

A079638 Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.

Original entry on oeis.org

1, 3, 1, 14, 9, 1, 90, 83, 18, 1, 744, 870, 275, 30, 1, 7560, 10474, 4275, 685, 45, 1, 91440, 143892, 70924, 14805, 1435, 63, 1, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1, 20603520, 38769840, 24870572, 7398972, 1151409, 98280, 4578

Offset: 1

Vladeta Jovovic, Jan 30 2003

Matrix product of unsigned Lah-triangle |A008297(n,k)| and Stirling1-triangle A008275(n,k) is unsigned Stirling1-triangle |A008275(n,k)|.
Also the Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016
Essentially the same as A131222. - Peter Bala, Feb 12 2022

			Triangle begins
     1;
     3,     1;
    14,     9,    1;
    90,    83,   18,   1;
   744,   870,  275,  30,  1;
  7560, 10474, 4275, 685, 45, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..1225 (rows n = 1..50, flattened).
William Keith, Rishi Nath, and James Sellers, On simultaneous (s, s+t, s+2t, ...)-core partitions, arXiv:2508.00074 [math.CO], 2025. See p. 3.

Cf. A002866 (row sums), A029767 (first column), A131222.

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> n!*(2^(n+1)-1), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, n! (2^(n + 1) - 1)], rows = 12];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

T(n, k) = Sum_{i=k..n} |A008297(n, i)| * |A008275(i, k)|.

E.g.f.: ((1-x)/(1-2*x))^y. - Vladeta Jovovic, Nov 22 2003

cp's OEIS Frontend

A178886 Divide, cell by cell, sequence A178802 by sequence A178888.

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A194649 Triangle of coefficients of a sequence of polynomials related to the enumeration of linear labeled rooted trees.

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Mathematica

Formula

A208057 Triangle by rows, generated from the odd integers and related to A000165.

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Maple

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Extensions

A302606 a(n) = n! * [x^n] exp(n*x)*arcsinh(x).

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A308939 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} (2*k - 1)!!*x^k/k!).

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Maple

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Formula

A001217 Sorted list of orders of Weyl groups of types A_n, B_n, D_n, E_n, F_4, G_2.

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A028371 a(n) = (2^(n-1)-1)!*2^((2^(n-1)-1)-1).

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Magma

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A052564 Expansion of e.g.f. x*(1-x)/(1-2*x).

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PARI

PARI

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A302606 a(n) = n! * [x^n] exp(nx)arcsinh(x).

A308939 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} (2k - 1)!!x^k/k!).

A052564 Expansion of e.g.f. x(1-x)/(1-2x).