A275422 -id:A275422 - cp's OEIS frontend

A190865 E.g.f. exp(x+x^3/6).

1, 1, 1, 2, 5, 11, 31, 106, 337, 1205, 5021, 20186, 86461, 417847, 1992355, 9860306, 53734241, 292816841, 1633818457, 9855157330, 59926837141, 370352343971, 2439935383271, 16283034762842, 109982177787505, 783404343570301, 5668314772422901, 41412522553362026

Offset: 0

Vladimir Kruchinin, May 22 2011

nonn

a(n) is the number of set partitions of {1,2,...,n} such that the size of each block divides 3. - Geoffrey Critzer, Sep 23 2011

			a(0) = 1 because (vacuously) all sizes of the blocks in the unique set partition of {} divide 3.
a(4) = 5 because there are 5 such set partitions of {1,2,3,4}: ({1},{2,3,4}) ({2},{1,3,4}) ({3},{1,2,4}) ({4},{1,2,3}) ({1},{2},{3},{4}).

Alois P. Heinz, Table of n, a(n) for n = 0..666

Cf. A001470.

Column k=3 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 3]))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 27 2016

Range[0, 25]! CoefficientList[Series[Exp[x + x^3/6] , {x, 0, 25}], x]

a(n):=n!*sum(1/((k)!*(n-3*k)!*6^(k)),k,0,n/3);

a(n) = n!*sum(k=0..n/3, 1/((k)!*(n-3*k)!*6^(k))), n>0, a(0)=1.
E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - x*(6+x^2)/(x*(6+x^2)+ 6*(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
Recurrence: 2*a(n) = 2*a(n-1) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Jun 27 2013
a(n) ~ n^(2*n/3) * exp(-2*n/3+(2*n)^(1/3)) / (sqrt(3)*2^(n/3)) * (1 - 2^(2/3)/(6*n^(1/3)) + 13*2^(1/3)/(36*n^(2/3))). - Vaclav Kotesovec, Jun 27 2013
a(n) = hypergeom([-n/3, (1 - n)/3, (2 - n)/3], [], -9/2). - Peter Luschny, Jun 04 2021

A275423 Number of set partitions of [n] such that five is a multiple of each block size.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 379, 1849, 9109, 37324, 128129, 507508, 3031393, 19609773, 108440893, 500515633, 2467616641, 17154715726, 134519207131, 927764339426, 5359830269641, 31580724696907, 248587878630807, 2259650025239257, 18541914182165557

Offset: 0

Alois P. Heinz, Jul 27 2016

nonn

			a(6) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.

Alois P. Heinz, Table of n, a(n) for n = 0..619
Wikipedia, Partition of a set

Column k=5 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 5]))
    end:
seq(a(n), n=0..30);
# second Maple program:
seq(simplify(hypergeom([-n/5, (1-n)/5, (2-n)/5, (3-n)/5, (4-n)/5], [], -625/24)), n = 0..28); # Karol A. Penson, Sep 14 2023.

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 5}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

a(n) = n!*sum(k=0, n\5, 1/5!^k*binomial(n-4*k, k)/(n-4*k)!); \\ Seiichi Manyama, Feb 26 2022

a(n) = if(n<5, 1, a(n-1)+binomial(n-1, 4)*a(n-5)); \\ Seiichi Manyama, Feb 26 2022

E.g.f.: exp(x+x^5/5!).
From Seiichi Manyama, Feb 26 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/5)} (1/5!)^k * binomial(n-4*k,k)/(n-4*k)!.
a(n) = a(n-1) + binomial(n-1,4) * a(n-5) for n > 4. (End)
a(n) = hypergeom([-n/5,(1-n)/5,(2-n)/5,(3-n)/5,(4-n)/5],[],-625/24). - Karol A. Penson, Sep 14 2023.

A275429 Number of set partitions of [n] such that n is a multiple of each block size.

Original entry on oeis.org

1, 1, 2, 2, 11, 2, 167, 2, 1500, 1206, 16175, 2, 3486584, 2, 3188421, 29226654, 772458367, 2, 130880325103, 2, 4173951684174, 623240762412, 644066092301, 2, 220076136813712815, 31580724696908, 538897996103277, 49207275464475052, 44147498142028751570, 2

Offset: 0

Alois P. Heinz, Jul 27 2016

nonn

			a(4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 2: 12345, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..587
Wikipedia, Partition of a set

Main diagonal of A275422.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, A(n-j, k)*binomial(n-1, j-1)), j=
      `if`(k=0, 1..n, numtheory[divisors](k))))
    end:
a:= n-> A(n$2):
seq(a(n), n=0..30);

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[If[j > n, 0, A[n - j, k]* Binomial[n - 1, j - 1]], {j, If[k == 0, Range[n], Divisors[k]]}]];
a[n_] := A[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

a(n) = n! * [x^n] exp(Sum_{d|n} x^d/d!) for n>0, a(0) = 1.
a(n) = A275422(n,n).
a(p) = 2 for p prime.

A190452 E.g.f. exp(x+x^2/2+x^4/24).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 106, 372, 1499, 6211, 28606, 135356, 697357, 3688049, 20935006, 121837276, 753159801, 4767863657, 31807384354, 217048147396, 1551200297291, 11327527814191, 86206555248122, 669666314150164, 5399592811359331, 44398500646885851

Offset: 0

Vladimir Kruchinin, May 24 2011

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..600

Column k=4 of A275422.

With[{nn=30},CoefficientList[Series[Exp[x+x^2/2+x^4/24],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 21 2012 *)

a(n):=n!*sum(sum(binomial(j,n-4*k+3*j)*12^(j-k)*binomial(k,j)*2^(-n+3*k-2*j),j,floor((4*k-n)/3),floor((4*k-n)/2))/k!,k,1,n);

N=33;  x='x+O('x^N);
egf=exp(x+x^2/2+x^4/4!);
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */

E.g.f.: exp(x+x^2/2+x^4/24).
a(n) = n!*sum(k=1..n, sum(j=floor((4*k-n)/3)..floor((4*k-n)/2), binomial(j,n-4*k+3*j)*12^(j-k)*binomial(k,j)*2^(-n+3*k-2*j))/k!), n>0, a(0)=1.
Recurrence: 6*a(n) = 6*a(n-1) + 6*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Oct 09 2013
a(n) ~ 1/2*exp((6*n)^(1/4) + sqrt(6*n)/2 - 3*n/4 - 3/4) * n^(3*n/4) * 6^(-n/4) * (1 + 3^(5/4)/(16*(2*n)^(3/4)) + 7*sqrt(3/2)/(8*sqrt(n)) - 3^(3/4)/(2*(2*n)^(1/4))). - Vaclav Kotesovec, Oct 09 2013

More terms from Harvey P. Dale, Jun 21 2012

A275425 Number of set partitions of [n] such that seven is a multiple of each block size.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 5149, 32176, 217361, 1186329, 5282785, 20004037, 66589681, 266164921, 2012163385, 18230119678, 137986473241, 849028203101, 4391743155801, 19722685412431, 98510163677641, 856572597342541, 9516244046786101

Offset: 0

Alois P. Heinz, Jul 27 2016

nonn

			a(8) = 9: 1234567|8, 1234568|7, 1234578|6, 1234678|5, 1235678|4, 1245678|3, 1345678|2, 1|2345678, 1|2|3|4|5|6|7|8.

Alois P. Heinz, Table of n, a(n) for n = 0..618
Wikipedia, Partition of a set

Column k=7 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 7]))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 7}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

a(n) = n!*sum(k=0, n\7, 1/7!^k*binomial(n-6*k, k)/(n-6*k)!); \\ Seiichi Manyama, Feb 26 2022

a(n) = if(n<7, 1, a(n-1)+binomial(n-1, 6)*a(n-7)); \\ Seiichi Manyama, Feb 26 2022

E.g.f.: exp(x+x^7/7!).
From Seiichi Manyama, Feb 26 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/7)} (1/7!)^k * binomial(n-6*k,k)/(n-6*k)!.
a(n) = a(n-1) + binomial(n-1,6) * a(n-7) for n > 6. (End)

A275424 Number of set partitions of [n] such that six is a multiple of each block size.

Original entry on oeis.org

1, 1, 2, 5, 14, 46, 167, 659, 2836, 13064, 64076, 333928, 1834438, 10592518, 64136528, 405519766, 2672202304, 18315499424, 130245129112, 959527765480, 7311915167696, 57536223460640, 466963917417152, 3904133599120624, 33583586584746728, 296948602314737576

Offset: 0

Alois P. Heinz, Jul 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..603
Wikipedia, Partition of a set

Column k=6 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 2, 3, 6]))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 2, 3, 6}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

E.g.f.: exp(x+x^2/2+x^3/6+x^6/6!).

A275426 Number of set partitions of [n] such that eight is a multiple of each block size.

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 106, 372, 1500, 6220, 28696, 136016, 702802, 3727946, 21253324, 124231096, 772458366, 4918962462, 33061094812, 227303566648, 1639389311906, 12082068225466, 92951836390172, 729991698024568, 5960615982017512, 49636995406898376

Offset: 0

Alois P. Heinz, Jul 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..604
Wikipedia, Partition of a set

Column k=8 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 2, 4, 8]))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 2, 4, 8}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

E.g.f.: exp(x+x^2/2+x^4/24+x^8/8!).

A275427 Number of set partitions of [n] such that nine is a multiple of each block size.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 31, 106, 337, 1206, 5031, 20241, 86901, 421422, 2014377, 10015461, 54946881, 301009311, 1692429867, 10319449158, 63321896601, 395830490301, 2648669976261, 17920165424382, 122976000215289, 894420751179276, 6596034524038701, 49207275464475051

Offset: 0

Alois P. Heinz, Jul 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..617
Wikipedia, Partition of a set

Column k=9 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 3, 9]))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 3, 9}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

E.g.f.: exp(x+x^3/6+x^9/9!).

A275428 Number of set partitions of [n] such that ten is a multiple of each block size.

Original entry on oeis.org

1, 1, 2, 4, 10, 27, 82, 274, 988, 3880, 16175, 72205, 340660, 1697060, 8906990, 48911059, 281486144, 1687198848, 10535484376, 68349098640, 459596780618, 3202506672898, 23052054364956, 171418420964352, 1314125642973640, 10375794542251692, 84315714183790792

Offset: 0

Alois P. Heinz, Jul 27 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..607
Wikipedia, Partition of a set

Column k=10 of A275422.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 2, 5, 10]))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 2, 5, 10}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

E.g.f.: exp(x+x^2/2+x^5/5!+x^10/10!).

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A190865 E.g.f. exp(x+x^3/6).

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A275423 Number of set partitions of [n] such that five is a multiple of each block size.

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A275429 Number of set partitions of [n] such that n is a multiple of each block size.

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A190452 E.g.f. exp(x+x^2/2+x^4/24).

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A275425 Number of set partitions of [n] such that seven is a multiple of each block size.

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A275424 Number of set partitions of [n] such that six is a multiple of each block size.

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A275426 Number of set partitions of [n] such that eight is a multiple of each block size.

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