A102233 -id:A102233 - cp's OEIS frontend

A032032 Number of ways to partition n labeled elements into sets of sizes of at least 2 and order the sets.

1, 0, 1, 1, 7, 21, 141, 743, 5699, 42241, 382153, 3586155, 38075247, 428102117, 5257446533, 68571316063, 959218642651, 14208251423433, 223310418094785, 3699854395380371, 64579372322979335, 1182959813115161773, 22708472725269799933, 455643187943171348103

Offset: 0

Christian G. Bower

nonn

From Dennis P. Walsh, Apr 15 2013: (Start)
With m = floor(n/2), a(n) is the number of ways to distribute n different toys to m numbered children such that each child receiving a toy gets at least two toys and, if child k gets no toys, then each child numbered higher than k also gets no toys.
a(n) = sum of n-th row of triangle A200091 for n >= 2. (End)

			For n=5, a(5)=21 since there are 21 toy distributions satisfying the conditions above. Denoting a distribution by |kid_1 toys|kid_2 toys|, we have the distributions
  |t1,t2,t3,t4,t5|_|, |t1,t2,t3|t4,t5|, |t1,t2,t4|t3,t5|, |t1,t2,t5|t3,t4|, |t1,t3,t4|t2,t5|, |t1,t3,t5|t2,t4|, |t1,t4,t5|t2,t3|, |t2,t3,t4|t1,t5|, |t2,t3,t5|t1,t4|, |t2,t4,t5|t1,t3|, |t3,t4,t5|t1,t2|, |t1,t2|t3,t4,t5|, |t1,t3|t2,t4,t5|, |t1,t4|t2,t3,t5|, |t1,t5|t2,t3,t4|, |t2,t3|t1,t4,t5|, |t2,t4|t1,t3,t5|, |t2,t5|t1,t3,t4|, |t3,t4|t1,t2,t5|, |t3,t5|t1,t2,t4|, and |t4,t5|,t1,t2,t3|. - _Dennis P. Walsh_, Apr 15 2013

Alois P. Heinz, Table of n, a(n) for n = 0..400
C. G. Bower, Transforms (2).
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see p. 245.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013 and J. Int. Seq. 17 (2014) #14.1.1.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Index entries for related partition-counting sequences

Cf. A102233, A232475.
Cf. column k=2 of A245732.
Cf. A200091.

spec := [ B, {B=Sequence(Set(Z,card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)];
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=2..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 29 2014

a[n_] := n! * Sum[ Binomial[k, j] * StirlingS2[n-k+j, j]*j! / (n-k+j)! * (-1)^(k-j), {k, 1, n}, {j, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Sep 05 2012, from given formula *)

x='x+O('x^66); Vec(serlaplace( 1/(2+x-exp(x)) ) ) \\ Joerg Arndt, Apr 16 2013

"AIJ" (ordered, indistinct, labeled) transform of 0, 1, 1, 1...
E.g.f.: 1/(2+x-exp(x)).
a(n) = n! * Sum_{k=1..n} Sum_{j=0..k} C(k,j) * Stirling2(n-k+j,j) * (j!/(n-k+j)!) *(-1)^(k-j); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
a(n) ~ n! / ((r-1)*(r-2)^(n+1)), where r = -LambertW(-1,-exp(-2)) = 3.14619322062... - Vaclav Kotesovec, Oct 08 2013
a(0) = 1; a(n) = Sum_{k=2..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
a(n) = Sum_{s in S_n^0} Product_{i=1..n} binomial(i,s(i)-1), where s ranges over the set S_n^0 of derangements of [n], i.e., the permutations of [n] without fixed points. - Jose A. Rodriguez, Feb 02 2021

A245732 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality >=k exists and a nonempty preimage of j implies that all i<=j have preimages with cardinality >=k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 27, 13, 1, 1, 256, 75, 7, 1, 1, 3125, 541, 21, 1, 1, 1, 46656, 4683, 141, 21, 1, 1, 1, 823543, 47293, 743, 71, 1, 1, 1, 1, 16777216, 545835, 5699, 183, 71, 1, 1, 1, 1, 387420489, 7087261, 42241, 2101, 253, 1, 1, 1, 1, 1

Offset: 0

Alois P. Heinz, Jul 30 2014

T(0,0) = 1 by convention.

In general, column k > 1 is asymptotic to n! / ((1+r^(k-1)/(k-1)!) * r^(n+1)), where r is the root of the equation 2 - exp(r) + Sum_{j=1..k-1} r^j/j! = 0. - Vaclav Kotesovec, Aug 02 2014

			Triangle T(n,k) begins:
0 :         1;
1 :         1,      1;
2 :         4,      3,    1;
3 :        27,     13,    1,   1;
4 :       256,     75,    7,   1,  1;
5 :      3125,    541,   21,   1,  1, 1;
6 :     46656,   4683,  141,  21,  1, 1, 1;
7 :    823543,  47293,  743,  71,  1, 1, 1, 1;
8 :  16777216, 545835, 5699, 183, 71, 1, 1, 1, 1;

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

Column k=0 gives A000312.
Columns k=1-10 give (for n>0): A000670, A032032, A102233, A232475, A245790, A245791, A245792, A245793, A245794, A245795.
T(2n,n) gives A244174(n) or 1+A007318(2n,n) = 1+A000984(n) for n>0.
Cf. A245733.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
T:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; T[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 05 2015, after Alois P. Heinz *)

E.g.f. (for column k > 0): 1/(2 -exp(x) +Sum_{j=1..k-1} x^j/j!) -1. - Vaclav Kotesovec, Aug 02 2014

A232475 Number of preferential arrangements of n labeled elements when at least k=4 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 71, 253, 673, 1585, 38149, 277707, 1402831, 5923503, 85577571, 937629969, 7475614341, 48939413477, 587610659505, 7906296686903, 87384175023995, 804959532778571, 9729015122635103, 144711323234918941, 2009073351016603121

Offset: 0

N. J. A. Sloane, Nov 27 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.

Cf. A032032, A102233.

Cf. column k=4 of A245732.

b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=4..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 29 2014

CoefficientList[Series[1/(2 + x - E^x + x^2/2 + x^3/6),{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Aug 02 2014 *)

E.g.f.: 1/(2 + x - exp(x) + x^2/2 + x^3/6). - Vaclav Kotesovec, Aug 02 2014
a(n) ~ n! / ((1+r^3/6) * r^(n+1)), where r = 1.97615974210650519398... is the root of the equation 2 + r - exp(r) + r^2/2 + r^3/6 = 0. - Vaclav Kotesovec, Aug 02 2014
a(0) = 1; a(n) = Sum_{k=4..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020

More terms from Alois P. Heinz, Jul 29 2014

A337058 E.g.f.: 1 / (2 + x^2/2 - exp(x)).

Original entry on oeis.org

1, 1, 2, 7, 33, 191, 1323, 10711, 99151, 1032385, 11943003, 151979213, 2109829857, 31730171539, 513903517585, 8917723105003, 165065061436755, 3246274767649637, 67598797715175999, 1485845872704318265, 34378343609138619685, 835190283258080561671

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

nonn

Cf. A000670, A032032, A097514, A102233, A337059.

nmax = 21; CoefficientList[Series[1/(2 + x^2/2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 21}]

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=3..n} binomial(n,k) * a(n-k).

A355284 Expansion of e.g.f. 1 / (1 + x + x^2/2 + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 2, 6, 24, 200, 1560, 12936, 130368, 1458432, 17623440, 233922480, 3376625472, 52382131776, 870882440064, 15459372915840, 291596692838400, 5824039155720192, 122814724467223296, 2726547887891407104, 63562453551393223680, 1552499303360183700480

Offset: 0

Ilya Gutkovskiy, Jun 26 2022

nonn

Cf. A007840, A038205, A102233, A226226, A355285.

nmax = 22; CoefficientList[Series[1/(1 + x + x^2/2 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(1/(1 + x + x^2/2 + log(1 - x)))) \\ Michel Marcus, Jun 27 2022

E.g.f.: 1 / (1 - Sum_{k>=3} x^k/k).

a(0) = 1; a(n) = Sum_{k=3..n} binomial(n,k) * (k-1)! * a(n-k).

A365911 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4k+3) / (4k+3)! ).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 20, 1, 0, 1680, 240, 1, 369600, 102960, 4160, 168168001, 76876800, 7743840, 137225153280, 93117024001, 17091609600, 182510023324320, 172080261401600, 49615854288001, 369403226582016000, 461748751736204400, 191552892427653120

Offset: 0

Seiichi Manyama, Sep 22 2023

Seiichi Manyama, Table of n, a(n) for n = 0..498

Cf. A102233, A332258, A365912.
Cf. A352429, A365917.
Cf. A307978.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(sinh(x)-sin(x))/2)))

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/4)} binomial(n,4*k+3) * a(n-4*k-3).

E.g.f.: 1 / ( 1 - (sinh(x) - sin(x))/2 ).

A341283 Number of ways to partition n labeled elements into sets of different sizes of at least 3.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 36, 57, 211, 331, 958, 29228, 64065, 294659, 1232479, 3549717, 11296603, 557617987, 1512758550, 8514685860, 41183585167, 251022906729, 838303110637, 4183056225010, 263978773601641, 887708421995331, 5813843897797861, 32212405278588967, 216518890998518716

Offset: 0

Ilya Gutkovskiy, Apr 28 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..699

Cf. A006505, A007837, A025148, A032311, A102233, A343319, A343542.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i+1)*binomial(n, i)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 28 2021

nmax = 29; CoefficientList[Series[Product[(1 + x^k/k!), {k, 3, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 2 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 29}]

E.g.f.: Product_{k>=3} (1 + x^k/k!).

A365912 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5k+3) / (5k+3)! ).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 20, 0, 1, 1680, 0, 330, 369600, 1, 180180, 168168000, 13990, 163363200, 137225088001, 39041010, 232792560000, 182509367449640, 118574979600, 494730748512001, 369398970833730090, 451334037000000, 1500683270499930350, 1080492079984609149000

Offset: 0

Seiichi Manyama, Sep 22 2023

Seiichi Manyama, Table of n, a(n) for n = 0..498

Cf. A102233, A332258, A365911.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\5, x^(5*k+3)/(5*k+3)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/5)} binomial(n,5*k+3) * a(n-5*k-3).

A245855 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 2.

Original entry on oeis.org

1, 0, 6, 20, 120, 672, 5516, 40140, 368640, 3521870, 37445298, 422339502, 5215454426, 68144100780, 954428684280, 14160968076584, 222769496190060, 3692874342747114, 64493471050666430, 1181830474135532130, 22692074431844298558, 455404848204906308984

Offset: 2

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..400

Column k=2 of A245733.

Cf. A032032, A102233, A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
a:= n-> b(n, 2) -b(n, 3):
seq(a(n), n=2..25);

E.g.f.: 1/(2-exp(x)+x) -1/(2-exp(x)+x+x^2/2).

a(n) = A032032(n) - A102233(n) = A245732(n,2) - A245732(n,3).

A245856 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 3.

Original entry on oeis.org

1, 0, 0, 20, 70, 112, 1848, 12840, 62700, 591800, 5484908, 40589276, 421291780, 4704380800, 46345716880, 533446290384, 6931113219780, 85313661653400, 1121432682942740, 16310909250477380, 237534778732260548, 3578871132644512672, 57980168196079811800

Offset: 3

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..400

Column k=3 of A245733.

Cf. A102233, A232475, A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
a:= n-> b(n, 3) -b(n, 4):
seq(a(n), n=3..30);

With[{nn=30},CoefficientList[Series[1/(2-Exp[x]+x+x^2/2)-1/(2-Exp[x]+ x+ x^2/2+ x^3/6),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 14 2016 *)

E.g.f.: 1/(2-exp(x)+x+x^2/2)-1/(2-exp(x)+x+x^2/2+x^3/6).

a(n) = A102233(n) - A232475(n) = A245732(n,3) - A245732(n,4).

cp's OEIS Frontend

A032032 Number of ways to partition n labeled elements into sets of sizes of at least 2 and order the sets.

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A245732 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality >=k exists and a nonempty preimage of j implies that all i<=j have preimages with cardinality >=k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A232475 Number of preferential arrangements of n labeled elements when at least k=4 elements per rank are required.

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A337058 E.g.f.: 1 / (2 + x^2/2 - exp(x)).

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A355284 Expansion of e.g.f. 1 / (1 + x + x^2/2 + log(1 - x)).

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A365911 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4*k+3) / (4*k+3)! ).

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A341283 Number of ways to partition n labeled elements into sets of different sizes of at least 3.

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Mathematica

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A365912 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5*k+3) / (5*k+3)! ).

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A365911 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(4k+3) / (4k+3)! ).

A365912 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5k+3) / (5k+3)! ).