A232475 -id:A232475 - 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

A102233 Number of preferential arrangements of n labeled elements when at least k=3 elements per rank are required.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 21, 71, 183, 2101, 13513, 64285, 629949, 5762615, 41992107, 427215283, 4789958371, 47283346849, 540921904725, 6980052633257, 85901272312905, 1129338979629643, 16398293425501375, 238339738265039119, 3588600147767147775, 58124879519314730741

Offset: 0

Thomas Wieder, Jan 01 2005

nonn

The labeled case for at least k=2 elements per rank is given by A032032 = Partition n labeled elements into sets of sizes of at least 2 and order the sets. The unlabeled case for at least k=3 elements per rank is given by A000930 = A Lamé sequence of higher order. The unlabeled case for at least k=2 elements per rank is given by A000045 = Fibonacci numbers.
With m = floor(n/3), 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 three toys and, if child k gets no toys, then each child numbered higher than k also gets no toys. Furthermore, a(n)= row sums of triangle A200092 for n>=3. - Dennis P. Walsh, Apr 15 2013
Row sums of triangle A200092. - Dennis P. Walsh, Apr 15 2013

			Let 1,2,3,4,5,6 denote six labeled elements. Let | denote a separation between two ranks. E.g., if elements 1,2 and 3 are on rank (also called level) one and elements 3,4 and 5 are on rank two, then we have the ranking 123|456.
For n=9 we have a(9)=2101 rankings. The order within a rank does not count. Six examples are: 123|456|789; 123456789; 12345|6789; 129|345678; 1235|46789; 789|123456.

Alois P. Heinz, Table of n, a(n) for n = 0..400
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
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.

Cf. A000640, A102232, A032032, A232475.

Cf. column k=3 of A245732.

seq (n! *coeff (series (1- (z^2-2*exp(z)+2+2*z) /(4-2*exp(z)+2*z+z^2), z=0, n+1), z, n), n=0..30);
with(combstruct): SeqSetL := [S, {S=Sequence(U), U=Set(Z, card >= 3)}, labeled]: seq(count(SeqSetL, size=j), j=0..23); # Zerinvary Lajos, Oct 19 2006
# third Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=3..n)) end:
a:= n-> n!*b(n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 29 2014

CoefficientList[Series[1-(x^2-2*E^x+2+2*x)/(4-2*E^x+2*x+x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2013 *)
b[n_] := b[n] = If[n==0, 1, Sum[b[n-j]/j!, {j, 3, n}]]; a[n_] := n!*b[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)

z='z+O('z^66); Vec(serlaplace( 1-(z^2-2*exp(z)+2+2*z) / (4-2*exp(z)+2*z+z^2) ) ) \\ Joerg Arndt, Apr 16 2013

E.g.f.: 1-(z^2-2*exp(z)+2+2*z)/(4-2*exp(z)+2*z+z^2).
a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i) /(i+k)!))); a(0)=1. - Vladimir Kruchinin, Feb 01 2011
a(n) ~ 2*n!/((2+r^2)*r^(n+1)), where r = 1.56811999239... is the root of the equation 4+2*r+r^2 = 2*exp(r). - Vaclav Kotesovec, Sep 29 2013
a(0) = 1; a(n) = Sum_{k=3..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Feb 09 2020
E.g.f.: 1/(2 + x + x^2/2 - exp(x)). - Christian Sievers, Oct 27 2024

a(0) changed to 1 at the suggestion of Zerinvary Lajos, Oct 26 2006

A337059 E.g.f.: 1 / (2 + x^3/6 - exp(x)).

Original entry on oeis.org

1, 1, 3, 12, 67, 461, 3823, 36933, 407963, 5068909, 69982083, 1062784273, 17607354955, 316012688213, 6108011298847, 126490611884013, 2794122884322635, 65578524701197341, 1629676370022564219, 42748628870263418761, 1180373377691425730235

Offset: 0

Ilya Gutkovskiy, Aug 13 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..426

Cf. A000670, A032032, A124504, A232475, A337058.

S:= series(1/(2+x^3/6-exp(x)),x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Aug 28 2020

nmax = 20; CoefficientList[Series[1/(2 + x^3/6 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = 1; a[n_] := a[n] = n (a[n - 1] + (n - 1) a[n - 2]/2) + Sum[Binomial[n, k] a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 20}]

a(0) = a(1) = 1; a(n) = n * (a(n-1) + (n-1) * a(n-2) / 2) + Sum_{k=4..n} binomial(n,k) * a(n-k).

A343319 Number of ways to partition n labeled elements into sets of different sizes of at least 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 127, 211, 793, 1288, 3719, 6007, 646439, 1467077, 7211843, 30123763, 91160937, 293184840, 1118980377, 110635063749, 319072758997, 1918239941962, 9518126978941, 58119248603131, 202992067559011, 1031021295578251, 4151156602678042, 650225250329137612

Offset: 0

Ilya Gutkovskiy, Apr 28 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700

Cf. A007837, A025149, A032311, A057837, A232475, A341283, 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, 4):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 28 2021

nmax = 30; CoefficientList[Series[Product[(1 + x^k/k!), {k, 4, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 3 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 30}]

E.g.f.: Product_{k>=4} (1 + x^k/k!).

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).

A245857 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 4.

Original entry on oeis.org

1, 0, 0, 0, 70, 252, 420, 660, 35640, 271700, 1389388, 5137860, 79463020, 905649500, 7336909980, 48400150764, 573924746400, 7735300382250, 85942063340210, 795156908528290, 9670781421636258, 143772253669334950, 1993964186469438950, 24015169625528033550

Offset: 4

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..400

Column k=4 of A245733.

Cf. A232475, A245790, 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, 4) -b(n, 5):
seq(a(n), n=4..30);

E.g.f.: 1/(1-Sum_{j>=4} x^j/j!) - 1/(1-Sum_{j>=5} x^j/j!).

a(n) = A232475(n) - A245790(n) = A245732(n,4) - A245732(n,5).

A365914 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5k+4) / (5k+4)! ).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 70, 1, 0, 0, 34650, 1430, 1, 0, 63063000, 5105100, 54740, 1, 305540235000, 40738698000, 1134117600, 1652090, 3246670537110001, 644180662125000, 33240837630000, 314389754250, 66475579247381221350, 18359921887357050001

Offset: 0

Seiichi Manyama, Sep 22 2023

Cf. A232475, A365913.

Cf. A365895.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\5, x^(5*k+4)/(5*k+4)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-4)/5)} binomial(n,5*k+4) * a(n-5*k-4).

A355285 Expansion of e.g.f. 1 / (1 + x + x^2/2 + x^3/3 + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 6, 24, 120, 720, 7560, 76608, 810432, 9141120, 118015920, 1666336320, 25211774016, 404932155264, 6951992261760, 127203705538560, 2467434718218240, 50477473338494976, 1086707769452699904, 24573149993692615680, 582367494447600583680, 14430857455114783119360

Offset: 0

Ilya Gutkovskiy, Jun 26 2022

nonn

Cf. A007840, A047865, A226226, A232475, A355284.

nmax = 23; CoefficientList[Series[1/(1 + x + x^2/2 + x^3/3 + 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, 4, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(1/(1 + x + x^2/2 + x^3/3 + log(1 - x)))) \\ Michel Marcus, Jun 27 2022

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

a(0) = 1; a(n) = Sum_{k=4..n} binomial(n,k) * (k-1)! * a(n-k).

A365913 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3k+4) / (3k+4)! ).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 70, 0, 1, 660, 34650, 1, 5434, 1351350, 63063001, 43656, 40694940, 6983776801, 305540584486, 1140183540, 550554404401, 77301682251156, 3246701716667574, 38582808660001, 13159690691494570, 1627974214800566490, 66478153367438069401

Offset: 0

Seiichi Manyama, Sep 22 2023

Cf. A232475, A365914.

Cf. A365894.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+4)/(3*k+4)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-4)/3)} binomial(n,3*k+4) * a(n-3*k-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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A102233 Number of preferential arrangements of n labeled elements when at least k=3 elements per rank are required.

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A337059 E.g.f.: 1 / (2 + x^3/6 - exp(x)).

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A343319 Number of ways to partition n labeled elements into sets of different sizes of at least 4.

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

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A245857 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 4.

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A365914 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5*k+4) / (5*k+4)! ).

A365914 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5k+4) / (5k+4)! ).

A365913 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3k+4) / (3k+4)! ).