A032032 -id:A032032 - cp's OEIS frontend

A343788 Number of ordered partitions of an n-set without blocks of size 5.

1, 1, 3, 13, 75, 540, 4671, 47125, 543371, 7048453, 101589591, 1610634433, 27856938387, 521953586233, 10532102378983, 227699187663961, 5250934660206219, 128659152359921997, 3337861722359261475, 91406502629924948053, 2634888477782107003707, 79751100251346500871481

Offset: 0

Ilya Gutkovskiy, Apr 29 2021

nonn

Cf. A000670, A032032, A337058, A337059, A343665, A343787, A343789, A343790, A343791, A343792, A343793.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=5, 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^5/5! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 5, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: 1 / (2 + x^5/5! - exp(x)).

A343789 Number of ordered partitions of an n-set without blocks of size 6.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4682, 47279, 545611, 7083565, 102182883, 1621425829, 28067555607, 526349480593, 10629883138059, 230009622202373, 5308749619032571, 130186940173803053, 3380385112758108315, 92650130825921846941, 2673020491585091254035, 80974418589343644492805

Offset: 0

Ilya Gutkovskiy, Apr 29 2021

nonn

Cf. A000670, A032032, A337058, A337059, A343666, A343787, A343788, A343790, A343791, A343792, A343793.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=6, 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^6/6! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 6, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

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

A343790 Number of ordered partitions of an n-set without blocks of size 7.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47292, 545819, 7086973, 102242283, 1622530933, 28089498891, 526813752973, 10640325166227, 230258631645913, 5315029292965675, 130353994525735677, 3385061859378821547, 92787606222541942477, 2677254928352340708075, 81110818086045534369661

Offset: 0

Ilya Gutkovskiy, Apr 29 2021

nonn

Cf. A000670, A032032, A337058, A337059, A343667, A343787, A343788, A343789, A343791, A343792, A343793.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=7, 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^7/7! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 7, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: 1 / (2 + x^7/7! - exp(x)).

A343791 Number of ordered partitions of an n-set without blocks of size 8.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545834, 7087243, 102247203, 1622625313, 28091415135, 526854986737, 10641264928479, 230281282588513, 5315605563021465, 130369438065006551, 3385496924633886429, 92800464391224494215, 2677652842774247060805, 81123688691904430522831

Offset: 0

Ilya Gutkovskiy, Apr 29 2021

nonn

Cf. A000670, A032032, A337058, A337059, A343668, A343787, A343788, A343789, A343790, A343792, A343793.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=8, 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^8/8! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 8, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: 1 / (2 + x^8/8! - exp(x)).

A343792 Number of ordered partitions of an n-set without blocks of size 9.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087260, 102247543, 1622632133, 28091557915, 526858128161, 10641337741219, 230283060907913, 5315651289289195, 130370674248854021, 3385532005327322503, 92801507648842580769, 2677685300845992661475, 81124743440296074264381

Offset: 0

Ilya Gutkovskiy, Apr 29 2021

nonn

Cf. A000670, A032032, A337058, A337059, A343669, A343787, A343788, A343789, A343790, A343791, A343793.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=9, 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^9/9! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 9, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: 1 / (2 + x^9/9! - exp(x)).

A343793 Number of ordered partitions of an n-set without blocks of size 10.

Original entry on oeis.org

1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247562, 1622632551, 28091567067, 526858335797, 10641342662135, 230283183134017, 5315654473869451, 130370761261559229, 3385534496252337939, 92801582269262225397, 2677687636903407184463, 81124819758167172293305

Offset: 0

Ilya Gutkovskiy, Apr 29 2021

nonn

Cf. A000670, A032032, A337058, A337059, A343671, A343787, A343788, A343789, A343790, A343791, A343792.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=10, 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^10/10! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 10, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: 1 / (2 + x^10/10! - exp(x)).

A050374 Number of ordered factorizations of n into composite factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 1, 0, 3, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 1, 1, 1, 0, 5, 1, 1, 1, 1, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 1, 5, 1, 1, 0, 1, 1, 1, 0, 7, 0, 1, 1, 1, 1, 1, 0, 5, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 1, 1, 1, 1, 10, 0, 1, 1, 4, 0, 1

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The Dirichlet inverse is given by A005171, all but the first term in A005171 turned negative. - R. J. Mathar, Jul 15 2010

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A002033, A002808, A005171, A050370-A050375, A000045, A002110, A032032.

read(transforms):
[1, seq(-A005171(n), n=2..100)] ;
a050374 := DIRICHLETi(%) ; # R. J. Mathar, May 26 2017

A050374(n) = if(1==n,n,sumdiv(n,d,if(dA050374(d),0))); \\ Antti Karttunen, Oct 20 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of composite numbers.
a(n) = A050375(A101296(n)). - R. J. Mathar, May 26 2017
For n >= 1, a(p^n) = A000045(n-1), for any prime p.
For n >= 0, a(A002110(n)) = A032032(n).

A343672 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

Original entry on oeis.org

1, 3, 19, 181, 2299, 36501, 695427, 15457709, 392672651, 11221959685, 356339728243, 12446649786429, 474273933636411, 19577992095770837, 870345573347448803, 41455153171478627533, 2106173029315813515883, 113694251997087087941925, 6498401704686168598548435, 392062852538564346207533789

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000670, A005493, A006155, A032032, A343673, A343674.

a[0] = 1; a[n_] := a[n] = 2 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(2 (1 - x) - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

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

a(n) ~ n! / (2*(1 + LambertW(exp(1)/2)) * (1 - LambertW(exp(1)/2))^(n+1)). - Vaclav Kotesovec, Jun 20 2022

A343673 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

Original entry on oeis.org

1, 4, 33, 409, 6759, 139621, 3460989, 100091335, 3308146179, 123005753041, 5081871122073, 230948185830187, 11449697796242319, 614944043618257237, 35568197580789653685, 2204201734650777596863, 145703352769994600516187, 10233323176300508748808921, 761004837938469796089586257

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000670, A005494, A006155, A032032, A343672, A343674.

a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 3 x - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (2 - 3*x - exp(x)).

a(n) ~ n! * 3^n / ((1 + LambertW(exp(2/3)/3)) * (2 - 3*LambertW(exp(2/3)/3))^(n+1)). - Vaclav Kotesovec, Jun 20 2022

A343674 a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

Original entry on oeis.org

1, 5, 51, 781, 15947, 407021, 12466251, 445452813, 18191122219, 835737327661, 42661645147403, 2395510523568845, 146739531459316587, 9737742346694258157, 695911661109898805323, 53286006304099668950413, 4352120920347139791200171, 377674509364714706139413933, 34702277449656625185428239755

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000670, A006155, A032032, A045379, A343672, A343673.

a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 (1 - 2 x) - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

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

a(n) ~ n! * 2^(n-1) / ((1 + LambertW(exp(1/2)/4)) * (1 - 2*LambertW(exp(1/2)/4))^(n+1)). - Vaclav Kotesovec, Jun 20 2022

cp's OEIS Frontend

A343788 Number of ordered partitions of an n-set without blocks of size 5.

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Maple

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A343789 Number of ordered partitions of an n-set without blocks of size 6.

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Maple

Mathematica

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A343790 Number of ordered partitions of an n-set without blocks of size 7.

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Maple

Mathematica

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A343791 Number of ordered partitions of an n-set without blocks of size 8.

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Maple

Mathematica

Formula

A343792 Number of ordered partitions of an n-set without blocks of size 9.

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Maple

Mathematica

Formula

A343793 Number of ordered partitions of an n-set without blocks of size 10.

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Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A050374 Number of ordered factorizations of n into composite factors.

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A343672 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

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Mathematica

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A343673 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

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