A000296 -id:A000296 - cp's OEIS frontend

A282988 Triangle of partitions of an n-set into boxes of size >= m.

1, 2, 1, 5, 1, 1, 15, 4, 1, 1, 52, 11, 1, 1, 1, 203, 41, 11, 1, 1, 1, 877, 162, 36, 1, 1, 1, 1, 4140, 715, 92, 36, 1, 1, 1, 1, 21147, 3425, 491, 127, 1, 1, 1, 1, 1, 115975, 17722, 2557, 337, 127, 1, 1, 1, 1, 1, 678570, 98253, 11353, 793, 463, 1, 1, 1, 1, 1, 1

Offset: 1

Vladimir Kruchinin, Feb 26 2017

			Triangle T(n,m) begins:
    1;
    2,   1;
    5,   1,   1;
   15,   4,   1,   1;
   52,  11,   1,   1,   1;
  203,  41,  11,   1,   1,   1;
  877, 162,  36,   1,   1,   1,   1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000110, A000296, A006505, A057837, A057814, A182931, A260878.

T:= proc(n, k) option remember; `if`(n=0, 1, add(
      T(n-j, k)*binomial(n-1, j-1), j=k..n))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 28 2017

T[n_, m_] := T[n, m] = Which[Or[n == m, n == 0], 1, m == 0, 0, True, Sum[Binomial[n - 1, i + m - 1] T[n - i - m, m], {i, 0, n - m}]]; Table[T[n, m], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, Feb 26 2017 *)

T(n,m):=if n=m or n=0 then 1 else if m=0 then 0 else sum(binomial(n-1, i+m-1)*T(n-i-m,m), i, 0, n-m);

T(n,m) = Sum_{i=0..n-m} C(n-1, i+m-1)*T(n-i-m, m).

E.g.f. m column of T(n,m) is exp(exp(x)-Sum_{k=0..m} 1/k!x^k).

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

A306418 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} requiring k steps of removing singletons and cyclical adjacency initiators until reaching a fixed point, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 3, 0, 1, 2, 12, 0, 0, 0, 12, 35, 5, 0, 0, 5, 56, 100, 42, 0, 0, 0, 14, 282, 343, 231, 7, 0, 0, 0, 66, 1406, 1476, 1088, 104, 0, 0, 0, 0, 307, 7592, 7383, 4929, 909, 27, 0, 0, 0, 0, 1554, 44227, 40514, 22950, 6240, 470, 20, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Feb 14 2019

See Callan's article for details on this transformation (SeparateIS).

			Triangle begins:
    1
    0    1
    0    2    0
    0    2    3    0
    1    2   12    0    0
    0   12   35    5    0    0
    5   56  100   42    0    0    0
   14  282  343  231    7    0    0    0
   66 1406 1476 1088  104    0    0    0    0
  307 7592 7383 4929  909   27    0    0    0    0

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

Row sums are A000110. First column is A324011.

Cf. A000126, A000296, A001610, A306416, A306417, A324012.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
qbj[stn_]:=With[{ini=Join@@Table[Select[s,If[#==Max@@Max@@@stn,MemberQ[s,First[Union@@stn]],MemberQ[s,(Union@@stn)[[Position[Union@@stn,#][[1,1]]+1]]]]&],{s,stn}],sng=Join@@Select[stn,Length[#]==1&]},DeleteCases[Table[Complement[s,Union[sng,ini]],{s,stn}],{}]];
Table[Length[Select[sps[Range[n]],Length[FixedPointList[qbj,#]]-2==k&]],{n,0,8},{k,0,n}]

A332248 Number of set partitions of [n] where all prime-indexed blocks are not singletons.

Original entry on oeis.org

1, 1, 1, 2, 5, 15, 60, 286, 1423, 7185, 37758, 212596, 1293577, 8415869, 57715274, 414520958, 3125102795, 24880061105, 209909409566, 1871945790360, 17503956383037, 169851122851049, 1694189515772750, 17248694322541778, 178473482993477591, 1873036127628583885

Offset: 0

Alois P. Heinz, Feb 12 2020

nonn

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 1|23.
a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
a(5) = 15: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1|23|45, 1|24|35, 1|25|34.

Alois P. Heinz, Table of n, a(n) for n = 0..576
Wikipedia, Partition of a set

Cf. A000040, A000110, A000296, A332398.

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, i+1)*
       binomial(n-1, j-1), j=`if`(isprime(i), 2, 1)..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..32);

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, i+1] Binomial[n-1, j-1], {j, If[PrimeQ[i], 2, 1], n}]];
a[n_] := b[n, 1];
a /@ Range[0, 32] (* Jean-François Alcover, May 08 2020, after Maple *)

A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

Original entry on oeis.org

1, 0, 1, 4, 14, 66, 397, 2626, 18797, 148238, 1281134, 11943790, 118998365, 1262189748, 14203022537, 168835162632, 2111832477426, 27708387132906, 380355066174121, 5449577398256414, 81316095965242989, 1261149374033472626, 20293627142875917978, 338263983223664609198

Offset: 0

Ilya Gutkovskiy, Sep 02 2021

nonn

Exponential transform of A000295.

Cf. A000295, A000296, A003725, A055882, A143405, A347434, A347435.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021

nmax = 23; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000295(k) * a(n-k).
a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A003725(k) * A143405(n-k).
a(n) ~ n^(n + 1/2) * (exp(exp(r)*(exp(r) - r - 1) - r/2 - n) / (r^(n + 1/2) * sqrt(2*exp(r)*(1 + 2*r) - (2 + r*(4 + r))))), where r = LambertW(n)/2 + (4 + LambertW(n)) * LambertW(n)^(3/2) / (8 * sqrt(n) * (1 + LambertW(n))). - Vaclav Kotesovec, Jul 07 2022

A355338 Expansion of e.g.f.: exp(exp(x) - x^2 - 1).

Original entry on oeis.org

1, 1, 0, -1, 3, 12, -7, -47, 332, 1347, -2105, -4200, 135457, 474697, -900832, 4682135, 126196787, 439488524, 233313817, 19129265609, 239146712732, 1104038984091, 5891696027079, 89831511761320, 911995655018817, 6253185308181553, 54873149768926624, 653039078246798383

Offset: 0

Vaclav Kotesovec, Jun 29 2022

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..578

Cf. A000296, A277381, A316778, A355337.

nmax = 30; CoefficientList[Series[Exp[Exp[x] - x^2 - 1], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x) - x^2 - 1))) \\ Michel Marcus, Jun 29 2022

a(n) ~ n^n * exp(n/LambertW(n) - LambertW(n)^2 - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^n).
a(n) ~ Bell(n) / exp(LambertW(n)^2).
a(0) = a(1) = 1; a(n) = -2 * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 29 2022

A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 3, 0, 0, 0, 1, 10, 0, 0, 0, 0, 1, 25, 15, 0, 0, 0, 0, 1, 56, 105, 0, 0, 0, 0, 0, 1, 119, 490, 105, 0, 0, 0, 0, 0, 1, 246, 1918, 1260, 0, 0, 0, 0, 0, 0, 1, 501, 6825, 9450, 945, 0, 0, 0, 0, 0, 0, 1, 1012, 22935, 56980, 17325, 0, 0, 0, 0, 0, 0

Offset: 0

Peter Luschny, Nov 25 2022

{{n, k}} are the number of k-quotient sets of an n-set having at least two elements in each equivalence class. This is the definition and notation (doubling the stacked delimiters of the Stirling set numbers) as given by Fekete (see link).
The formal definition expresses the second order Stirling set numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of second kind' used elsewhere should be dropped in favor of the more systematic one used here.
Also the Bell transform of sign(n) for n >= 0. For the definition of the Bell transform see A264428.

			Triangle T(n, k) starts:
[0] 1;
[1] 0, 0;
[2] 0, 1,   0;
[3] 0, 1,   0,    0;
[4] 0, 1,   3,    0,    0;
[5] 0, 1,  10,    0,    0,  0;
[6] 0, 1,  25,   15,    0,  0,  0;
[7] 0, 1,  56,  105,    0,  0,  0,  0;
[8] 0, 1, 119,  490,  105,  0,  0,  0,  0;
[9] 0, 1, 246, 1918, 1260,  0,  0,  0,  0,  0;

Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, 2nd ed. 1994, thirty-fourth printing 2022.

Antal E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

A008299 is an irregular subtriangle with more information.
A358622 (second order Stirling cycle numbers).
Cf. A000296 (row sums), alternating row sums (apart from sign): A000587, A293037, and A014182.
Cf. A048993, A201637, A264428, A269939, A340264.

T := (n, k) -> add(binomial(n, k - j)*Stirling2(n - k + j, j)*(-1)^(k - j),
j = 0..k): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Using the e.g.f.:
egf := exp(z*(exp(t) - t - 1)): ser := series(egf, t, 12):
seq(print(seq(n!*coeff(coeff(ser, t, n), z, k), k = 0..n)), n = 0..9);
# Using second order Eulerian numbers:
A358623 := proc(n, k) if n = 0 then return 1 fi;
add(binomial(j, n - 2*k)*combinat:-eulerian2(n - k, n - k - j - 1), j = 0..n-k-1)
end: seq(seq(A358623(n, k), k = 0..n), n = 0..11);

# recursion over rows
from functools import cache
@cache
def StirlingSetOrd2(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 0]
    rov: list[int] = StirlingSetOrd2(n - 2)
    row: list[int] = StirlingSetOrd2(n - 1) + [0]
    for k in range(1, n // 2 + 1):
        row[k] = (n - 1) * rov[k - 1] + k * row[k]
    return row
for n in range(9): print(StirlingSetOrd2(n))
# Alternative, using function BellMatrix from A264428.
def f(k: int) -> int:
    return 1 if k > 0 else 0
print(BellMatrix(f, 9))

T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(j, k - j)*<>, where <> denote the second order Eulerian numbers (extending Knuth's notation).
T(n, k) = n!*[z^k][t^n] exp(z*(exp(t) - t - 1)).
T(n, k) = Sum_{j=0..k} (-1)^(k - j)*binomial(n, k - j)*{n - k + j, j}, where {n, k} denotes the Stirling set numbers.
T(n, k) = (n - 1) * T(n-2, k-1) + k * T(n-1, k) with suitable boundary conditions.
T(n + k, k) = A269939(n, k), which might be called the Ward set numbers.

A365338 Expansion of e.g.f. exp( Sum_{k>=0} x^(3k+2) / (3k+2)! ).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 15, 21, 106, 378, 1116, 6931, 20196, 136500, 558559, 3103191, 18524418, 90852217, 661904622, 3569749659, 25657118047, 171935108541, 1149965503347, 9166884110308, 62758177512570, 525194926694256, 4072360499426311, 33295056117878646

Offset: 0

Seiichi Manyama, Sep 22 2023

nonn

Cf. A000296, A365419.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(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-1,3*k+1) * a(n-3*k-2).

A365419 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+2) / (5k+2)! ).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 15, 1, 105, 36, 945, 990, 10396, 25740, 136942, 675675, 2238405, 18378361, 50490765, 523833819, 1665638325, 15790609395, 71976549646, 515107015830, 3524757498023, 19222751356875, 181500385651325, 888496839591076, 9603885700913400

Offset: 0

Seiichi Manyama, Sep 22 2023

nonn

Cf. A000296, A365338.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(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-1,5*k+1) * a(n-5*k-2).

A367890 Expansion of e.g.f. exp(3*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 3, 3, 30, 93, 633, 3342, 22809, 156063, 1183872, 9453711, 80455125, 721576560, 6809391111, 67332650007, 695777512638, 7493572404345, 83926492573341, 975467527353750, 11744536832206149, 146234590864310019, 1880198749437144456, 24928860500681953683

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A027710, A194689, A346738, A355253, A355254, A367888, A367891.

nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - 3 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-3)^n / k!.
a(0) = 1; a(n) = 3 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A027710(k).

cp's OEIS Frontend

A282988 Triangle of partitions of an n-set into boxes of size >= m.

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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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A306418 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} requiring k steps of removing singletons and cyclical adjacency initiators until reaching a fixed point, n >= 0, 0 <= k <= n.

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A332248 Number of set partitions of [n] where all prime-indexed blocks are not singletons.

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A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

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A355338 Expansion of e.g.f.: exp(exp(x) - x^2 - 1).

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A358623 Regular triangle read by rows. T(n, k) = {{n, k}}, where {{n, k}} are the second order Stirling set numbers (or second order Stirling numbers). T(n, k) for 0 <= k <= n.

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A365338 Expansion of e.g.f. exp( Sum_{k>=0} x^(3*k+2) / (3*k+2)! ).

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A365338 Expansion of e.g.f. exp( Sum_{k>=0} x^(3k+2) / (3k+2)! ).

A365419 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+2) / (5k+2)! ).