A032032 -id:A032032 - cp's OEIS frontend

A052841 Expansion of e.g.f.: 1/(exp(x)*(2-exp(x))).

1, 0, 2, 6, 38, 270, 2342, 23646, 272918, 3543630, 51123782, 811316286, 14045783798, 263429174190, 5320671485222, 115141595488926, 2657827340990678, 65185383514567950, 1692767331628422662, 46400793659664205566, 1338843898122192101558, 40562412499252036940910

Offset: 0

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

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A005359(n)=[0,2,0,24,0,720,...] is a(n)=[0,2,6,38,270,...].
Stirling transform of -(-1)^n*A052657(n-1)=[0,0,2,-6,48,-240,...] is a(n-1)=[0,0,2,6,38,270,...].
Stirling transform of -(-1)^n*A052558(n-1)=[1,-1,4,-12,72,-360,...] is a(n-1)=[1,0,2,6,38,270,...].
Stirling transform of 2*A052591(n)=[2,4,24,96,...] is a(n+1)=[2,6,38,270,...].
(End)
Also the central moments of a Geometric(1/2) random variable (for example the number of coin tosses until the first head). - Svante Janson, Dec 10 2012
Also the number of ordered set partitions of {1..n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). - Gus Wiseman, Feb 13 2019
Also the number of ordered set partitions of {1..n} with an even number of blocks. - Geoffrey Critzer, Jul 04 2020

			From _Gus Wiseman_, Feb 13 2019: (Start)
The a(4) = 38 ordered set partitions with no cyclical adjacencies:
  {{1}{2}{3}{4}}  {{1}{24}{3}}  {{13}{24}}
  {{1}{2}{4}{3}}  {{1}{3}{24}}  {{24}{13}}
  {{1}{3}{2}{4}}  {{13}{2}{4}}
  {{1}{3}{4}{2}}  {{13}{4}{2}}
  {{1}{4}{2}{3}}  {{2}{13}{4}}
  {{1}{4}{3}{2}}  {{2}{4}{13}}
  {{2}{1}{3}{4}}  {{24}{1}{3}}
  {{2}{1}{4}{3}}  {{24}{3}{1}}
  {{2}{3}{1}{4}}  {{3}{1}{24}}
  {{2}{3}{4}{1}}  {{3}{24}{1}}
  {{2}{4}{1}{3}}  {{4}{13}{2}}
  {{2}{4}{3}{1}}  {{4}{2}{13}}
  {{3}{1}{2}{4}}
  {{3}{1}{4}{2}}
  {{3}{2}{1}{4}}
  {{3}{2}{4}{1}}
  {{3}{4}{1}{2}}
  {{3}{4}{2}{1}}
  {{4}{1}{2}{3}}
  {{4}{1}{3}{2}}
  {{4}{2}{1}{3}}
  {{4}{2}{3}{1}}
  {{4}{3}{1}{2}}
  {{4}{3}{2}{1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
C. G. Bower, Transforms (2)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 808
Svante Janson, Euler-Frobenius numbers and rounding, preprint arXiv:1305.3512 [math.PR], 2013.
Lukas Spiegelhofer, A lower bound for Cusick's conjecture on the digits of n+t, arXiv:1910.13170 [math.NT], 2019.

Main diagonal of A122101.
Inverse binomial transform of A000670.
Cf. A000126, A000296, A000670, A001610, A005359, A032032, A052558.
Cf. A052591, A052657, A052841, A124323, A156365, A169985, A173018.
Cf. A187784, A324011, A344037, A346208.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024

spec := [S,{B=Prod(C,C),C=Set(Z,1 <= card),S=Sequence(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
P := proc(n,x) option remember; if n = 0 then 1 else
(n*x+2*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x); expand(%) fi end:
A052841 := n -> subs(x=2, P(n,x)):
seq(A052841(n), n=0..21); # Peter Luschny, Mar 07 2014
h := n -> add(combinat:-eulerian1(n, k)*2^k, k=0..n):
a := n -> (h(n)+(-1)^n)/2: seq(a(n), n=0..21); # Peter Luschny, Sep 19 2015
b := proc(n, m) option remember; if n = 0 then 1 else
     (m - 1)*b(n - 1, m) + (m + 1)*b(n - 1, m + 1) fi end:
a := n -> b(n, 0): seq(a(n), n = 0..21); # Peter Luschny, Jun 23 2023

a[n_] := If[n == 0, 1, (PolyLog[-n, 1/2]/2 + (-1)^n)/2]; (* or *)
a[n_] := HurwitzLerchPhi[1/2, -n, -1]/2; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 19 2016, after Vladeta Jovovic *)
With[{nn=30},CoefficientList[Series[1/(Exp[x](2-Exp[x])),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Apr 08 2019 *)

a(n)=if(n<0,0,n!*polcoeff(subst(1/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{a(n)=polcoeff(sum(m=0,n,(2*m)!*x^(2*m)/prod(k=1,2*m,1-k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

def A052841_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(x)) ).egf_to_ogf().list()
A052841_list(40) # G. C. Greubel, Jun 11 2024

O.g.f.: Sum_{n>=0} (2*n)! * x^(2*n) / Product_{k=1..2*n} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n) = (A000670(n) + (-1)^n)/2 = Sum_{k>=0} (k-1)^n/2^(k+1). - Vladeta Jovovic, Feb 02 2003
Also, a(n) = Sum_{k=0..[n/2]} (2k)!*Stirling2(n, 2k). - Ralf Stephan, May 23 2004
a(n) = D^n*(1/(1-x^2)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A005649. - Peter Bala, Nov 25 2011
E.g.f.: 1/(2*G(0)), where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k*(k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
a(n) ~ n!/(4*(log(2))^(n+1)). - Vaclav Kotesovec, Aug 10 2013
a(n) = (h(n)+(-1)^n)/2 where h(n) = Sum_{k=0..n} E(n,k)*2^k and E(n,k) the Eulerian numbers A173018 (see also A156365). - Peter Luschny, Sep 19 2015
a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 11 2020

Edited by N. J. A. Sloane, Sep 06 2013

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

A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 4, 4, 6, 0, 1, 11, 20, 10, 10, 0, 1, 41, 66, 60, 20, 15, 0, 1, 162, 287, 231, 140, 35, 21, 0, 1, 715, 1296, 1148, 616, 280, 56, 28, 0, 1, 3425, 6435, 5832, 3444, 1386, 504, 84, 36, 0, 1, 17722, 34250, 32175, 19440, 8610, 2772, 840, 120, 45, 0, 1

Offset: 0

Emeric Deutsch, Oct 28 2006

Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k) = binomial(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n) = A052889(n) = n*B(n-1), where B(q) are the Bell numbers (A000110).
Exponential Riordan array [exp(exp(x)-1-x),x]. - Paul Barry, Apr 23 2009
Sum_{k=0..n} T(n,k)*2^k = A000110(n+1) is the number of binary relations on an n-set that are both symmetric and transitive. - Geoffrey Critzer, Jul 25 2014
Also the number of set partitions of {1, ..., n} with k cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). Unlike A250104, we count {{1}} as having 1 cyclical adjacency. - Gus Wiseman, Feb 13 2019

			T(4,2)=6 because we have 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34 (if we take {1,2,3,4} as our 4-set).
Triangle starts:
     1
     0    1
     1    0    1
     1    3    0    1
     4    4    6    0    1
    11   20   10   10    0    1
    41   66   60   20   15    0    1
   162  287  231  140   35   21    0    1
   715 1296 1148  616  280   56   28    0    1
  3425 6435 5832 3444 1386  504   84   36    0    1
From _Gus Wiseman_, Feb 13 2019: (Start)
Row n = 5 counts the following set partitions by number of singletons:
  {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
  {{12}{34}}  {{123}{4}}  {{1}{23}{4}}
  {{13}{24}}  {{124}{3}}  {{12}{3}{4}}
  {{14}{23}}  {{134}{2}}  {{1}{24}{3}}
                          {{13}{2}{4}}
                          {{14}{2}{3}}
... and the following set partitions by number of cyclical adjacencies:
  {{13}{24}}      {{1}{2}{34}}  {{1}{234}}  {{1234}}
  {{1}{24}{3}}    {{1}{23}{4}}  {{12}{34}}
  {{13}{2}{4}}    {{12}{3}{4}}  {{123}{4}}
  {{1}{2}{3}{4}}  {{14}{2}{3}}  {{124}{3}}
                                {{134}{2}}
                                {{14}{23}}
(End)
From _Paul Barry_, Apr 23 2009: (Start)
Production matrix is
0, 1,
1, 0, 1,
1, 2, 0, 1,
1, 3, 3, 0, 1,
1, 4, 6, 4, 0, 1,
1, 5, 10, 10, 5, 0, 1,
1, 6, 15, 20, 15, 6, 0, 1,
1, 7, 21, 35, 35, 21, 7, 0, 1,
1, 8, 28, 56, 70, 56, 28, 8, 0, 1 (End)

Alois P. Heinz, Rows n = 0..140, flattened
David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.
T. Mansour, A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.

Cf. A000110, A052889, A124324.
A250104 is an essentially identical triangle, differing only in row 1.
For columns see A000296, A250105, A250106, A250107.
Cf. A000126, A001610, A032032, A052841, A066982, A080107, A169985, A187784, A324011, A324014, A324015.

G:=exp(exp(z)-1+(t-1)*z): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form
# Program from R. J. Mathar, Jan 22 2015:
A124323 := proc(n,k)
    binomial(n,k)*A000296(n-k) ;
end proc:

Flatten[CoefficientList[Range[0,10]! CoefficientList[Series[Exp[x y] Exp[Exp[x] - x - 1], {x, 0,10}], x], y]] (* Geoffrey Critzer, Nov 24 2011 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Count[#,{}]==k&]],{n,0,9},{k,0,n}] (* _Gus Wiseman, Feb 13 2019 *)

T(n,k) = binomial(n,k)*[(-1)^(n-k)+sum((-1)^(j-1)*B(n-k-j), j=1..n-k)], where B(q) are the Bell numbers (A000110).
E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)*z).
G.f.: 1/(1-xy-x^2/(1-xy-x-2x^2/(1-xy-2x-3x^2/(1-xy-3x-4x^2/(1-... (continued fraction). - Paul Barry, Apr 23 2009

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

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

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).

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).

A200091 The number of ways of putting n labeled items into k labeled boxes so that each box receives at least 2 objects.

Original entry on oeis.org

1, 1, 1, 6, 1, 20, 1, 50, 90, 1, 112, 630, 1, 238, 2940, 2520, 1, 492, 11508, 30240, 1, 1002, 40950, 226800, 113400, 1, 2024, 137610, 1367520, 2079000, 1, 4070, 445896, 7271880, 22869000, 7484400, 1, 8164, 1410552, 35692800, 196396200, 194594400, 1, 16354

Offset: 2

Peter Bala, Dec 04 2011

Equivalently, the number of ordered set partitions of the set [n] into k blocks of size at least two. When the boxes are unlabeled or the set partitions unordered we obtain A008299.
The number of doubly-surjective functions f:[n]->[k], where a doubly-surjective function f has pre-image sets of size at least 2 for each element of the codomain. Also, the number of ways to distribute n different toys to k different children so that each child gets at least two toys. - Dennis P. Walsh, Apr 09 2013
T(n,k) is the number of chains 0 = x_0 < x_1 < ... < x_k = 1 in the Boolean lattice of rank n, such that x_i is not covered by x_(i+1) for all i. - Geoffrey Critzer, Jul 15 2018

			Table begins
  n |k=1   2     3     4
----+-------------------
  2 |  1
  3 |  1
  4 |  1   6
  5 |  1  20
  6 |  1  50    90
  7 |  1 112   630
  8 |  1 238  2940  2520
  9 |  1 492 11508 30240
  ...
T(4,2) = 6: The arrangements of 4 objects into 2 boxes { } and [ ] so that each box contains at least 2 items are {1,2}[3,4], {1,3}[2,4], {2,3}[1,4] and the 3 other possibilities where the contents of a pair of boxes are swapped.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, page 100-109.

Muniru A Asiru, Table of n, a(n) for n = 2..626
Marko R. Riedel, Count by PIE and symbolic combinatorics, Math Stackexchange, March 2022.
Dennis Walsh, Notes on doubly-surjective finite functions

T(n,2) = A052515(n), T(n,3) = A224541(n), T(n,4) = A224542(n).
Cf. A032032 (row sums).
Cf. A008299, A019538, A200092.

Flat(List([2..14],n->List([1..Int(n/2)],k->Sum([0..k],j->(-1)^j*Binomial(k,j)*(Sum([0..j],i->Binomial(j,i)*(Binomial(n,i)*Factorial(i)*(k-j)^(n-i)))))))); # Muniru A Asiru, Jul 17 2018

seq(seq(eval(diff((exp(x)-x-1)^k,x$n),x=0),k=1..floor(n/2)),n=2..20); # Dennis P. Walsh, Apr 09 2013
T := proc(n,k) local r; k!* add(binomial(n,r)*(-1)^r*Stirling2(n-r,k-r), r=0..min(n,k)); end; # Marko Riedel, Mar 25 2022

t[n_, k_] := k! * Sum[ (-1)^i*Binomial[n, i] * Sum[ (-1)^j*(k - i - j)^(n - i) / (j!*(k - i - j)!), {j, 0, k - i}], {i, 0, k}]; Table[ t[n, k], {n, 2, 14}, {k, 1, n/2}] // Flatten (* Jean-François Alcover, Apr 10 2013 *)

E.g.f. with additional 1: 1/(1-t*(exp(x)-1-x)) = 1 + t*x^2/2! + t*x^3/3! + (t+6*t^2)*x^4/4! + ....
E.g.f. (k fixed): (exp(x)-x-1)^k. - Dennis P. Walsh, Apr 09 2013
Recurrence relation: T(n+1,k) = k*(T(n,k) + n*T(n-1,k-1)). T(n,k) = k!*A008299(n,k).
T(n,k+j) = Sum_{i=0..n} C(n,i)*T(i,k)*T(n-i,j). - Dennis P. Walsh, Apr 09 2013
T(n,k) = Sum_{j=0..k} (-1)^j*C(k,j)*(Sum_{i=0..j} C(j,i)*C(n,i)*i!*(k-j)^(n-i)) for 1 <= k <= n/2 and n >= 2. - Dennis P. Walsh, Apr 10 2013
T(n,k) = k!*Sum_{r=0..min(n,k)} binomial(n,r)*(-1)^r*Stirling2(n-r, k-r). - Marko Riedel, Mar 25 2022

A306417 Number of self-conjugate set partitions of {1, ..., n}.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 7, 7, 46, 39, 321

Offset: 0

Gus Wiseman, Feb 14 2019

This sequence counts set partitions fixed under Callan's conjugation operation.

			The  a(3) = 1 through a(7) = 7 self-conjugate set partitions:
  {{12}{3}}  {{13}{24}}  {{123}{4}{5}}  {{135}{246}}    {{13}{246}{57}}
                         {{13}{2}{45}}  {{124}{35}{6}}  {{15}{246}{37}}
                                        {{13}{25}{46}}  {{1234}{5}{6}{7}}
                                        {{14}{2}{356}}  {{124}{3}{56}{7}}
                                        {{14}{236}{5}}  {{134}{2}{5}{67}}
                                        {{14}{25}{36}}  {{14}{2}{3}{567}}
                                        {{145}{26}{3}}  {{14}{23}{57}{6}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A001610, A032032, A052841, A080107, A169985, A306416, A324011, A324012.

A343787 Number of ordered partitions of an n-set without blocks of size 4.

Original entry on oeis.org

1, 1, 3, 13, 74, 531, 4563, 45753, 524345, 6760039, 96837333, 1525909903, 26230304235, 488472319271, 9796281435125, 210496933103743, 4824574494068495, 117490079786298641, 3029472152485535343, 82454398253005541089, 2362311059301928969755, 71063998308414194250901

Offset: 0

Ilya Gutkovskiy, Apr 29 2021

nonn

Cf. A000670, A032032, A337058, A337059, A343664, A343788, A343789, A343790, A343791, A343792, A343793.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=4, 0, a(n-j)*binomial(n, j)), j=1..n))
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 29 2021

nmax = 21; CoefficientList[Series[1/(2 + x^4/4! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 4, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: 1 / (2 + x^4/4! - exp(x)).

cp's OEIS Frontend

A052841 Expansion of e.g.f.: 1/(exp(x)*(2-exp(x))).

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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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A124323 Triangle read by rows: T(n,k) is the number of partitions of an n-set having k singleton blocks (0<=k<=n).

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

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

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

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