A276921 -id:A276921 - cp's OEIS frontend

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

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

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)

			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)

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.
Laura Gellert and Raman Sanyal, On degree sequences of undirected, directed, and bidirected graphs, arXiv preprint arXiv:1512.08448 [math.CO], 2015.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
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

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.

[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

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

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 *)

Vec(serlaplace((2/(2-2*x-x^2) + O(x^30)))) \\ Michel Marcus, Nov 02 2015

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

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

A276922 Number T(n,k) of ordered set partitions of [n] where the maximal block size equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 42, 8, 1, 0, 120, 330, 80, 10, 1, 0, 720, 2970, 860, 120, 12, 1, 0, 5040, 30240, 10290, 1540, 168, 14, 1, 0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1, 0, 362880, 4377240, 1977360, 326970, 38808, 3696, 288, 18, 1

Offset: 0

Alois P. Heinz, Sep 22 2016

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2,      1;
  0,     6,      6,      1;
  0,    24,     42,      8,     1;
  0,   120,    330,     80,    10,    1;
  0,   720,   2970,    860,   120,   12,   1;
  0,  5040,  30240,  10290,  1540,  168,  14,  1;
  0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1;
  ...

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

Columns k=0-10 give: A000007, A000142 (for n>0), A320758, A320759, A320760, A320761, A320762, A320763, A320764, A320765, A320766.
Row sums give A000670.
T(2n,n) gives A276923.
Cf. A080510, A276921.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
       A(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; T[n_, k_] :=  A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 11 2017, translated from Maple *)

E.g.f. for column k>0: 1/(1-Sum_{i=1..k} x^i/i!) - 1/(1-Sum_{i=1..k-1} x^i/i!).

T(n,k) = A276921(n,k) - A276921(n,k-1) for k>0. T(n,0) = A000007(0).

A189886 a(n) is the number of compositions of the set {1, 2, ..., n} into blocks, each of size 1, 2 or 3 (n >= 0).

Original entry on oeis.org

1, 1, 3, 13, 74, 530, 4550, 45570, 521640, 6717480, 96117000, 1512819000, 25975395600, 483169486800, 9678799930800, 207733600074000, 4755768505488000, 115681418156304000, 2979408725813520000, 80998627977002736000, 2317937034142810080000, 69649003197501567840000, 2192459412316607834400000, 72152830779716793506400000, 2477756318984329979756160000

Offset: 0

Adi Dani, Apr 29 2011

nonn

Sequences of sets, each set having no more than 3 elements.

			a(3) = 13 because all compositions of set {a,b,c} into blocks of size 1, 2, or 3 are:
1: ({a,b,c}),
2: ({a},{b,c}),
3: ({b,c},{a}),
4: ({b},{a,c}),
5: ({a,c},{b}),
6: ({c},{a,b}),
7: ({a,b},{c}),
8: ({a},{b},{c}),
9: ({a},{c},{b}),
10: ({b},{a},{c}),
11: ({b},{c},{a}),
12: ({c},{a},{b}),
13: ({c},{b},{a}).

Alois P. Heinz, Table of n, a(n) for n = 0..424
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)
Adi Dani, Compositions and partitions of sets

Cf. A144422, A000670, A001680.

Column k=3 of A276921.

A189886 := proc(n) local m, j; add(add(2^(2*m-n-j)*3^(j-m)*n!
*binomial(m,j)*binomial(j,2*j-(3*m-n)),j=0..3*m-n),m=0..n) end:
seq(A189886(n),n=0..24); # Peter Luschny, May 02 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 3)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 22 2016
# third Maple program:
a:= n-> n! * (<<0|1|0>, <0|0|1>, <1/6|1/2|1>>^n)[3, 3]:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 22 2016

Table[Sum[n!/(2^(n+j-2m)3^(m-j))*Binomial[m,j]*Binomial[j,n+2j-3m], {m,0,n},{j,0,3m-n}],{n,0,15}]

a(n)=sum(m=0,n, sum(j=0,3*m-n, n!/(2^(n+j-2*m) *3^(m-j)) *binomial(m,j) *binomial(j,n+2*j-3*m))); /* Joerg Arndt, May 03 2011 */

a(n) = sum(m=0..n, sum(j=0..3*m-n, n!/(2^(n+j-2*m) * 3^(m-j)) * C(m,j) * C(j,n+2*j-3*m))) where C(n,k) is the binomial coefficient.
a(n) = n * a(n-1) + n*(n-1)/2 * a(n-2) + n*(n-1)*(n-2)/6 * a(n-3). - Istvan Mezo, Jun 06 2013
E.g.f.: 1/(1 - x - x^2/2 - x^3/6). -  Geoffrey Critzer, Dec 04 2012

A276924 Number of ordered set partitions of [n] with at most four elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 540, 4670, 47110, 543130, 7044450, 101519250, 1609319250, 27830729850, 521397676800, 10519576867800, 227400111939000, 5243385642495000, 128458209887007000, 3332234177825553000, 91241046790816923000, 2629791992312269785000

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=4 of A276921.

Cf. A001681.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 4)))
    end:
seq(a(n), n=0..25);
# second Maple program:
a:= n-> n!*(<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1/24|1/6|1/2|1>>^n)[4, 4]:
seq(a(n), n=0..25);

max = 20; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 4}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..4} x^i/i!).

A276925 Number of ordered set partitions of [n] with at most five elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4682, 47278, 545594, 7083258, 102177222, 1621316466, 28065324210, 526301293518, 10628781887724, 229983021824556, 5308071700475544, 130168746864660504, 3379871981604782664, 92634950510491052664, 2672550322299614660904

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=5 of A276921.

Cf. A110038.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 5)))
    end:
seq(a(n), n=0..25);

max = 25; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 5}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..5} x^i/i!).

A276926 Number of ordered set partitions of [n] with at most six elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47292, 545818, 7086954, 102241902, 1622523210, 28089336198, 526810157874, 10640241569958, 230256584914356, 5314976561846952, 130352566702702344, 3385021286975255928, 92786398312428612792, 2677217312112863784264

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=6 of A276921.

Cf. A148092.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 6)))
    end:
seq(a(n), n=0..25);

max = 25; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 6}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..6} x^i/i!).

A276927 Number of ordered set partitions of [n] with at most seven elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545834, 7087242, 102247182, 1622624850, 28091404902, 526854753282, 10641259374174, 230281144233426, 5315601950221992, 130369339178641992, 3385494089758915992, 92800379366660198376, 2677650178353887869704

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=7 of A276921.

Cf. A229224.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 7)))
    end:
seq(a(n), n=0..25);

max = 25; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 7}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..7} x^i/i!).

A276928 Number of ordered set partitions of [n] with at most eight elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087260, 102247542, 1622632110, 28091557362, 526858114926, 10641337416138, 230283052622766, 5315651069181882, 130370668142722722, 3385531828379161890, 92801502294634265418, 2677685131818279016434

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=8 of A276921.

Cf. A229225.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 8)))
    end:
seq(a(n), n=0..25);

max = 25; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 8}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..8} x^i/i!).

A276929 Number of ordered set partitions of [n] with at most nine elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247562, 1622632550, 28091567042, 526858335146, 10641342645362, 230283182692706, 5315654461874042, 130370760923004602, 3385534486308684710, 92801581965119911026, 2677687627216659136794

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=9 of A276921.

Cf. A229226.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 9)))
    end:
seq(a(n), n=0..25);

max = 25; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 9}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..9} x^i/i!).

A276930 Number of ordered set partitions of [n] with at most ten elements per block.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, 1622632572, 28091567570, 526858347730, 10641342953670, 230283190536542, 5315654669985946, 130370766690581274, 3385534653313192094, 92801587015186096762, 2677687786557636155446

Offset: 0

Alois P. Heinz, Sep 22 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..424

Column k=10 of A276921.

Cf. A229227.

a:= proc(n) option remember; `if`(n=0, 1, add(
       a(n-i)*binomial(n, i), i=1..min(n, 10)))
    end:
seq(a(n), n=0..25);

max = 25; CoefficientList[1/(1-Sum[x^i/i!, {i, 1, 10}]) + O[x]^(max+1), x]* Range[0, max]! (* Jean-François Alcover, May 24 2018 *)

E.g.f.: 1/(1-Sum_{i=1..10} x^i/i!).

cp's OEIS Frontend

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

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A276922 Number T(n,k) of ordered set partitions of [n] where the maximal block size equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A189886 a(n) is the number of compositions of the set {1, 2, ..., n} into blocks, each of size 1, 2 or 3 (n >= 0).

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A276924 Number of ordered set partitions of [n] with at most four elements per block.

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A276925 Number of ordered set partitions of [n] with at most five elements per block.

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A276926 Number of ordered set partitions of [n] with at most six elements per block.

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A276927 Number of ordered set partitions of [n] with at most seven elements per block.

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