A080599 Expansion of e.g.f.: 2/(2-2*x-x^2).

1, 1, 3, 12, 66, 450, 3690, 35280, 385560, 4740120, 64751400, 972972000, 15949256400, 283232149200, 5416632421200, 110988861984000, 2425817682288000, 56333385828720000, 1385151050307024000, 35950878932544576000, 982196278209226080000, 28175806418228108640000

Offset

0,3

Comments

Number of ordered partitions of {1,..,n} with at most 2 elements per block. - Bob Proctor, Apr 18 2005

In other words, number of preferential arrangements of n things (see A000670) in which each clump has size 1 or 2. - N. J. A. Sloane, Apr 13 2014

Recurrences (of the hypergeometric type of the Jovovic formula) mean: multiplying the sequence vector from the left with the associated matrix of the recurrence coefficients (here: an infinite lower triangular matrix with the natural numbers in the main diagonal and the triangular series in the subdiagonal) recovers the sequence up to an index shift. In that sense, this sequence here and many other sequences of the OEIS are eigensequences. - Gary W. Adamson, Feb 14 2011

Number of intervals in the weak (Bruhat) order of S_n that are Boolean algebras. - Richard Stanley, May 09 2011

a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A000085, A005442 and A052585. - Peter Bala, Dec 07 2011

From Gus Wiseman, Jul 04 2020: (Start)

Also the number of (1,1,1)-avoiding or cubefree sequences of length n covering an initial interval of positive integers. For example, the a(0) = 1 through a(3) = 12 sequences are:

() (1) (11) (112)

(12) (121)

(21) (122)

(123)

(132)

(211)

(212)

(213)

(221)

(231)

(312)

(321)

(End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 10.

Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Martínez Mori, Boolean intervals in the weak order of S_n, arXiv:2306.14734 [math.CO], 2023.

L. Gellert and R. Sanyal, On degree sequences of undirected, directed, and bidirected graphs, arXiv preprint arXiv:1512.08448 [math.CO], 2015.

Harri Hakula, Helmut Harbrecht, Vesa Kaarnioja, Frances Y. Kuo, and Ian H. Sloan, Uncertainty quantification for random domains using periodic random variables, arXiv:2210.17329 [math.NA], 2022.

Dixy Msapato, Counting the number of tau-exceptional sequences over Nakayama algebras, arXiv:2002.12194 [math.RT], 2020.

Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Index entries for related partition-counting sequences

Formula

a(n) = n*a(n-1) + (n*(n-1)/2)*a(n-2). - Vladeta Jovovic, Aug 22 2003

E.g.f.: 1/(1-x-x^2/2). - Richard Stanley, May 09 2011

a(n) ~ n!*((1+sqrt(3))/2)^(n+1)/sqrt(3). - Vaclav Kotesovec, Oct 13 2012

a(n) = n!*((1+sqrt(3))^(n+1) - (1-sqrt(3))^(n+1))/(2^(n+1)*sqrt(3)). - Vladimir Reshetnikov, Oct 31 2015

a(n) = A090932(n) * A002530(n+1). - Robert Israel, Nov 01 2015

Example

From Gus Wiseman, Jul 04 2020: (Start)
The a(0) = 1 through a(3) = 12 ordered set partitions with block sizes <= 2 are:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}
             {{1},{2}}  {{1,2},{3}}
             {{2},{1}}  {{1,3},{2}}
                        {{2},{1,3}}
                        {{2,3},{1}}
                        {{3},{1,2}}
                        {{1},{2},{3}}
                        {{1},{3},{2}}
                        {{2},{1},{3}}
                        {{2},{3},{1}}
                        {{3},{1},{2}}
                        {{3},{2},{1}}
(End)

Maple

a:= n-> n! *(Matrix([[1, 1], [1/2, 0]])^n)[1, 1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 01 2009
a:= gfun:-rectoproc({a(n) = n*a(n-1)+(n*(n-1)/2)*a(n-2), a(0)=1, a(1)=1}, a(n), remember):
seq(a(n), n=0..40); # Robert Israel, Nov 01 2015

Mathematica

Table[n!*SeriesCoefficient[-2/(-2+2*x+x^2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *)
Round@Table[n! ((1+Sqrt[3])^(n+1) - (1-Sqrt[3])^(n+1))/(2^(n+1) Sqrt[3]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

Prog

(PARI) Vec(serlaplace((2/(2-2*x-x^2) + O(x^30)))) \\ Michel Marcus, Nov 02 2015
(Magma) [n le 2 select 1 else (n-1)*Self(n-1) + Binomial(n-1, 2)*Self(n-2): n in [1..31]]; // G. C. Greubel, Jan 31 2023
(SageMath)
A002605=BinaryRecurrenceSequence(2, 2, 0, 1)
def A080599(n): return factorial(n)*A002605(n+1)/2^n
[A080599(n) for n in range(41)] # G. C. Greubel, Jan 31 2023

Crossrefs

Cf. A000085, A000670, A002530, A002605, A005442, A052585, A052611.

Cf. A090932, A102726, A106356, A232464, A333755, A335455, A335456.

Column k=2 of A276921.

Cubefree numbers are A004709.

(1,1)-avoiding patterns are A000142.

(1,1,1)-avoiding compositions are A232432.

(1,1,1)-matching patterns are A335508.

(1,1,1)-avoiding permutations of prime indices are A335511.

(1,1,1)-avoiding compositions are ranked by A335513.

(1,1,1,1)-avoiding patterns are A189886.

Sequence in context: A199746 A293302 A248871 * A349581 A120575 A009362

Adjacent sequences: A080596 A080597 A080598 * A080600 A080601 A080602

Keyword

nonn,eigen

Author

Detlef Pauly (dettodet(AT)yahoo.de), Feb 24 2003

Status

approved