A032032 -id:A032032 - cp's OEIS frontend

A187784 Triangular array read by rows: T(n,k) is the number of ordered set partitions of {1,2,...,n} with exactly k singletons, n>=0, 0<=k<=n.

1, 0, 1, 1, 0, 2, 1, 6, 0, 6, 7, 8, 36, 0, 24, 21, 100, 60, 240, 0, 120, 141, 372, 1170, 480, 1800, 0, 720, 743, 3584, 5166, 13440, 4200, 15120, 0, 5040, 5699, 22864, 67368, 68544, 159600, 40320, 141120, 0, 40320, 42241, 225684, 502200, 1161216, 922320, 1995840, 423360, 1451520, 0, 362880

Offset: 0

Geoffrey Critzer, Jan 05 2013

A singleton is a set that contains exactly one element.
Column for k=0 is A032032.
Row sums are A000670.
Main diagonal is A000142.

			:    1;
:    0,     1;
:    1,     0,     2;
:    1,     6,     0,     6;
:    7,     8,    36,     0,     24;
:   21,   100,    60,   240,      0,   120;
:  141,   372,  1170,   480,   1800,     0,    720;
:  743,  3584,  5166, 13440,   4200, 15120,      0, 5040;
: 5699, 22864, 67368, 68544, 159600, 40320, 141120,    0, 40320;

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

Cf. A000142, A000670, A032032.

with(combinat):
b:= proc(n, i, p) option remember; `if`(n=0, p!,
      `if`(i<2, 0, add(multinomial(n, n-i*j, i$j)
      *b(n-i*j, i-1, p+j)/j!, j=0..n/i)))
    end:
T:= (n, k)-> binomial(n, k)*b(n-k$2, k):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Sep 06 2015

nn=8;Range[0,nn]!CoefficientList[Series[1/(2-Exp[x]+x-y x),{x,0,nn}],{x,y}]//Grid

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

A332258 E.g.f.: 1 / (1 + x - sinh(x)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 20, 1, 112, 1681, 492, 27721, 371624, 319177, 13461604, 171387217, 319071456, 11466038689, 143550642140, 484491620089, 15758152572952, 199089883272217, 1077471975974484, 32827750137627457, 427744154995090256, 3385134777669637681

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Number of labeled ordered partitions of an n-set into odd parts > 1.

Cf. A006154, A009227, A032032.

nmax = 25; CoefficientList[Series[1/(1 + x - Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k - 1] a[n - 2 k + 1], {k, 2, Ceiling[n/2]}]; Table[a[n], {n, 0, 25}]

seq(n)={Vec(serlaplace(1 / (1 + x - sinh(x + O(x*x^n)))))} \\ Andrew Howroyd, Feb 08 2020

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

a(n) ~ n! / ((cosh(r) - 1) * r^(n+1)), where r = 1.72911689821437486498840709347... is the root of the equation 1 + r - sinh(r) = 0. - Vaclav Kotesovec, Feb 08 2020

A367838 Expansion of e.g.f. 1/(2 + x - exp(2*x)).

Original entry on oeis.org

1, 1, 6, 38, 344, 3832, 51408, 803952, 14371456, 289005440, 6457624832, 158719896832, 4255775425536, 123619815742464, 3867071262472192, 129610289219999744, 4633674344869756928, 176011269522607144960, 7079115958438736363520

Offset: 0

Seiichi Manyama, Dec 02 2023

nonn

Cf. A032032, A367839, A367840.

Cf. A367835.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-i*v[i]+sum(j=1, i, 2^j*binomial(i, j)*v[i-j+1])); v;

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

A367840 Expansion of e.g.f. 1/(2 + x - exp(4*x)).

Original entry on oeis.org

1, 3, 34, 514, 10456, 265704, 8103120, 288302480, 11722944896, 536262671488, 27256865214208, 1523936708699904, 92949383868668928, 6141694449341637632, 437033351625771001856, 33319937543640487708672, 2709708041047388536274944

Offset: 0

Seiichi Manyama, Dec 02 2023

nonn

Cf. A032032, A367838, A367839.

Cf. A367786, A367837.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-i*v[i]+sum(j=1, i, 4^j*binomial(i, j)*v[i-j+1])); v;

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

A245854 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 1.

Original entry on oeis.org

1, 2, 12, 68, 520, 4542, 46550, 540136, 7045020, 101865410, 1619046418, 28053492348, 526430246264, 10636085523910, 230214619661790, 5314695463338704, 130356558777712468, 3385311352838750538, 92797887464933030762, 2677623216872061223780, 81123642038690958720048

Offset: 1

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..400

Column k=1 of A245733.

Cf. A000670, A032032, 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, 1) -b(n, 2):
seq(a(n), n=1..25);

With[{nn=30},CoefficientList[Series[1/(2-Exp[x])-1/(2-Exp[x]+x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 29 2024 *)

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

a(n) = A000670(n) - A032032(n) = A245732(n,1) - A245732(n,2).

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).

A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 26, 84, 950, 6000, 62522, 556116, 6259598, 69319848, 874356338, 11384093196, 161462123894, 2397736692144, 37994808171962, 631767062124564, 11088109048500158, 203828700127054008, 3928762035148317314, 79079452776283889820, 1661265965479375937030, 36332908076071038467520, 826376466514358722894154

Offset: 0

Gus Wiseman, Feb 14 2019

nonn

			The a(4) = 2 ordered set partitions are: {{1,3},{2,4}}, {{2,4},{1,3}}.

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

Cf. A000110, A000126, A000296, A000670, A001610, A032032 (adjacencies allowed), A052841 (singletons allowed), A124323, A169985, A306417, A324011 (orderless case), A324012, A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Length[stn]!,{stn,Select[sps[Range[n]],And[Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]}],{n,0,10}]

a(12)-a(26) from Alois P. Heinz, Feb 14 2019

A365908 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3k+2) / (3k+2)! ).

Original entry on oeis.org

1, 0, 1, 0, 6, 1, 90, 42, 2521, 2268, 113742, 166321, 7543206, 16218930, 691242553, 2044833336, 83708046246, 324830941345, 12951273345282, 63596620804122, 2493395633726425, 15062005915534116, 584749646165678622, 4247497704703187089, 164155618660742879022

Offset: 0

Seiichi Manyama, Sep 22 2023

Cf. A032032, A365909.

Cf. A365338.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\3, x^(3*k+2)/(3*k+2)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-2)/3)} binomial(n,3*k+2) * a(n-3*k-2).

A365909 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5k+2) / (5k+2)! ).

Original entry on oeis.org

1, 0, 1, 0, 6, 0, 90, 1, 2520, 72, 113400, 5940, 7484401, 617760, 681084014, 81081000, 81730916280, 13232419201, 12505020896160, 2639867731518, 2376002176470000, 633568693965570, 548870403972290401, 180329793856173720, 151492831528555510516

Offset: 0

Seiichi Manyama, Sep 22 2023

Cf. A032032, A365908.

Cf. A365419.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=0, N\5, x^(5*k+2)/(5*k+2)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-2)/5)} binomial(n,5*k+2) * a(n-5*k-2).

A367839 Expansion of e.g.f. 1/(2 + x - exp(3*x)).

Original entry on oeis.org

1, 2, 17, 183, 2679, 48903, 1071621, 27394965, 800378019, 26307021483, 960739737777, 38595129840369, 1691405818822719, 80301792637126791, 4105701241574252445, 224912022483008478141, 13142159127790633537947, 815924005186398537216483

Offset: 0

Seiichi Manyama, Dec 02 2023

nonn

Cf. A032032, A367838, A367840.

Cf. A367785, A367836.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-i*v[i]+sum(j=1, i, 3^j*binomial(i, j)*v[i-j+1])); v;

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

cp's OEIS Frontend

A187784 Triangular array read by rows: T(n,k) is the number of ordered set partitions of {1,2,...,n} with exactly k singletons, n>=0, 0<=k<=n.

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A332258 E.g.f.: 1 / (1 + x - sinh(x)).

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A367838 Expansion of e.g.f. 1/(2 + x - exp(2*x)).

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A367840 Expansion of e.g.f. 1/(2 + x - exp(4*x)).

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A245854 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 1.

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Maple

Mathematica

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

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A306416 Number of ordered set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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A365908 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3*k+2) / (3*k+2)! ).

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PARI

Formula

A365909 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5*k+2) / (5*k+2)! ).

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A365908 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(3k+2) / (3k+2)! ).

A365909 Expansion of e.g.f. 1 / ( 1 - Sum_{k>=0} x^(5k+2) / (5k+2)! ).