A124504 -id:A124504 - cp's OEIS frontend

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

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).

A343664 Number of partitions of an n-set without blocks of size 4.

Original entry on oeis.org

1, 1, 2, 5, 14, 47, 173, 702, 3125, 14910, 76495, 418035, 2418397, 14791597, 95093612, 641094695, 4521228732, 33250447919, 254585084539, 2024995604762, 16702070759557, 142642458681486, 1259387604241013, 11479967000116911, 107910143688962037, 1044735841257587203, 10407104137208385924

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..581

Cf. A000110, A000296, A027338, A097514, A124504, A343665, A343666, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..26);  # Alois P. Heinz, Apr 25 2021

nmax = 26; CoefficientList[Series[Exp[Exp[x] - 1 - x^4/4!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 4 k]/((n - 4 k)! k! (4!)^k), {k, 0, Floor[n/4]}], {n, 0, 26}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 4, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 26}]

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

a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k * Bell(n-4*k) / ((n-4*k)! * k! * (4!)^k).

A343665 Number of partitions of an n-set without blocks of size 5.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 197, 835, 3860, 19257, 102997, 586170, 3535645, 22496437, 150454918, 1054235150, 7718958995, 58905868192, 467530598983, 3851775136517, 32881385742460, 290387471713872, 2649226725182823, 24934118754400767, 241809265181914545, 2413608066257526577

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027339, A097514, A124504, A343664, A343666, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=5, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 5 k]/((n - 5 k)! k! (5!)^k), {k, 0, Floor[n/5]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 5, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^5/5!).

a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * Bell(n-5*k) / ((n-5*k)! * k! * (5!)^k).

A343666 Number of partitions of an n-set without blocks of size 6.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 870, 4084, 20727, 112825, 654546, 4026487, 26145511, 178550986, 1278168860, 9564026947, 74615547996, 605593775899, 5103054929621, 44564754448972, 402677613100491, 3759094788129312, 36205919126040190, 359340174509911325, 3670825700549853053

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027340, A097514, A124504, A343664, A343665, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=6, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^6/6!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 6 k]/((n - 6 k)! k! (6!)^k), {k, 0, Floor[n/6]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 6, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^6/6!).

a(n) = n! * Sum_{k=0..floor(n/6)} (-1)^k * Bell(n-6*k) / ((n-6*k)! * k! * (6!)^k).

A343667 Number of partitions of an n-set without blocks of size 7.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 876, 4132, 21075, 115375, 673620, 4172413, 27296089, 187891174, 1356343385, 10238632307, 80615222404, 660560758879, 5621465069117, 49594663447612, 452846969975391, 4273130715906123, 41612346388251187, 417668648929556073, 4315893703814296053

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027341, A097514, A124504, A343664, A343665, A343666, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=7, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^7/7!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 7 k]/((n - 7 k)! k! (7!)^k), {k, 0, Floor[n/7]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 7, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^7/7!).

a(n) = n! * Sum_{k=0..floor(n/7)} (-1)^k * Bell(n-7*k) / ((n-7*k)! * k! * (7!)^k).

A343668 Number of partitions of an n-set without blocks of size 8.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4139, 21138, 115885, 677745, 4206172, 27577513, 190289713, 1377315050, 10426866782, 82350895629, 677003941219, 5781485704892, 51193839084907, 469251258854001, 4445769329586348, 43475305461354931, 438270620701587657, 4549243731200717053

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027342, A097514, A124504, A343664, A343665, A343666, A343667, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=8, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^8/8!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 8 k]/((n - 8 k)! k! (8!)^k), {k, 0, Floor[n/8]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 8, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^8/8!).

a(n) = n! * Sum_{k=0..floor(n/8)} (-1)^k * Bell(n-8*k) / ((n-8*k)! * k! * (8!)^k).

A343669 Number of partitions of an n-set without blocks of size 9.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115965, 678460, 4212497, 27633712, 190795218, 1381942530, 10470109267, 82764226404, 681048663329, 5822029128397, 51610194855972, 473631475252041, 4492967510009533, 43996047374513046, 444151309687221889, 4617189912288741028

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027343, A097514, A124504, A343664, A343665, A343666, A343667, A343668.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=9, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 25 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^9/9!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 9 k]/((n - 9 k)! k! (9!)^k), {k, 0, Floor[n/9]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 9, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^9/9!).

a(n) = n! * Sum_{k=0..floor(n/9)} (-1)^k * Bell(n-9*k) / ((n-9*k)! * k! * (9!)^k).

A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).

Original entry on oeis.org

1, 1, 2, 4, 1, 11, 4, 32, 20, 113, 80, 10, 422, 385, 70, 1788, 1792, 560, 8015, 9492, 3360, 280, 39435, 50640, 23100, 2800, 204910, 295020, 147840, 30800, 1144377, 1763300, 1044120, 246400, 15400, 6722107, 11278410, 7241520, 2202200, 200200, 41877722

Offset: 0

Emeric Deutsch, Nov 14 2006

Row n contains 1+floor(n/3) terms. Row sums yield the Bell numbers (A000110). T(n,0)=A124504(n). Sum(k*T(n,k), k=0..floor(n/3))=A105480(n+1).

			T(4,1)=4 because we have 1|234, 134|2, 124|3 and 123|4.
Triangle starts:
    1;
    1;
    2;
    4,   1;
   11,   4;
   32,  20;
  113,  80, 10;
  422, 385, 70;
  ...

Alois P. Heinz, Rows n = 0..250, flattened

Cf. A000110, A124504, A105480, A355144.

T(3n,n) gives A025035.

G:=exp(exp(z)-1+(t-1)*z^3/6): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1)*`if`(i=3, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Mar 08 2015

nn = 8; k = 3; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 + (y - 1) x^k/k!], {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 26 2012 *)

E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)z^3/6).

A328153 Number of set partitions of [n] such that at least one of the block sizes is 3.

Original entry on oeis.org

0, 0, 0, 1, 4, 20, 90, 455, 2352, 13132, 76540, 473660, 3069220, 20922330, 149021600, 1109629885, 8604815520, 69437698160, 581661169640, 5051885815603, 45411759404560, 421977921782270, 4047693372023070, 40034523497947132, 407818256494533984, 4274309903558446900

Offset: 0

Alois P. Heinz, Oct 05 2019

nonn

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

Column k=3 of A327884.

Cf. A000110, A124504, A328155.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(n, 0)-b(n, 3):
seq(a(n), n=0..27);

b[n_, k_] := b[n, k] = If[n==0, 1, Sum[If[j==k, 0, b[n-j, k] Binomial[n-1, j-1]], {j, 1, n}]];
a[n_] := b[n, 0] - b[n, 3];
a /@ Range[0, 27] (* Jean-François Alcover, May 02 2020, after Maple *)

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

a(n) = A000110(n) - A124504(n).

A361489 Expansion of e.g.f. exp(exp(x) - 1 + x^3/6).

Original entry on oeis.org

1, 1, 2, 6, 19, 72, 313, 1472, 7612, 42679, 255515, 1632710, 11065057, 79065807, 594174922, 4679473130, 38500353667, 330172915164, 2944613004359, 27253908250340, 261328607398332, 2591724561444621, 26545170005412613, 280411070646125638

Offset: 0

Seiichi Manyama, Mar 14 2023

Cf. A124504, A360991.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(exp(x)-1+x^3/6)))

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

cp's OEIS Frontend

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

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A343664 Number of partitions of an n-set without blocks of size 4.

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Maple

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A343665 Number of partitions of an n-set without blocks of size 5.

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Maple

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A343666 Number of partitions of an n-set without blocks of size 6.

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Maple

Mathematica

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A343667 Number of partitions of an n-set without blocks of size 7.

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Maple

Mathematica

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A343668 Number of partitions of an n-set without blocks of size 8.

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Maple

Mathematica

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A343669 Number of partitions of an n-set without blocks of size 9.

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Maple

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A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).

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Maple

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Formula

A328153 Number of set partitions of [n] such that at least one of the block sizes is 3.

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