A324162 -id:A324162 - cp's OEIS frontend

A327504 Number of set partitions of [n] where each subset is again partitioned into three nonempty subsets.

1, 0, 0, 1, 6, 25, 100, 511, 3626, 29765, 250200, 2146771, 19575446, 195336505, 2124840900, 24646324431, 299803782466, 3809251939245, 50698296967600, 708349718638891, 10372758309704686, 158546862369781985, 2519789706502636700, 41545703617137280551

Offset: 0

Alois P. Heinz, Sep 14 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..501
Wikipedia, Partition of a set

Column k=3 of A324162.

Cf. A346894.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 3), j=3..n))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j -1] StirlingS2[j, 3], {j, 3, n}]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/(6^k*k!)); \\ Seiichi Manyama, May 07 2022

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

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(6^k * k!). - Seiichi Manyama, May 07 2022

A060311 Expansion of e.g.f. exp((exp(x)-1)^2/2).

Original entry on oeis.org

1, 0, 1, 3, 10, 45, 241, 1428, 9325, 67035, 524926, 4429953, 40010785, 384853560, 3925008361, 42270555603, 478998800290, 5693742545445, 70804642315921, 918928774274028, 12419848913448565, 174467677050577515, 2542777209440690806, 38388037137038323353

Offset: 0

Vladeta Jovovic, Mar 27 2001

nonn

After the first term, this is the Stirling transform of the sequence of moments of the standard normal (or "Gaussian") probability distribution. It is not itself a moment sequence of any probability distribution. - Michael Hardy (hardy(AT)math.umn.edu), May 29 2005

a(n) is the number of simple labeled graphs on n nodes in which each component is a complete bipartite graph. - Geoffrey Critzer, Dec 03 2011

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, Ex. 3.3.5b.

Alois P. Heinz, Table of n, a(n) for n = 0..518 (first 101 terms from Harry J. Smith)
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A324162.

Cf. A052859, A330047.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 2), j=2..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2019

a = Exp[x] - 1; Range[0, 20]! CoefficientList[Series[Exp[a^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Dec 03 2011 *)

a(n)=if(n<0, 0, n!*polcoeff( exp((exp(x+x*O(x^n))-1)^2/2), n)) /* Michael Somos, Jun 01 2005 */

{ for (n=0, 100, write("b060311.txt", n, " ", n!*polcoeff(exp((exp(x + x*O(x^n)) - 1)^2/2), n)); ) } \\ Harry J. Smith, Jul 03 2009

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/(2^k*k!)); \\ Seiichi Manyama, May 07 2022

E.g.f. A(x) = B(exp(x)-1) where B(x)=exp(x^2/2) is e.g.f. of A001147(2n), hence a(n) is the Stirling transform of A001147(2n). - Michael Somos, Jun 01 2005
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ exp(1/2*(exp(r)-1)^2 - n) * n^(n+1/2) / (r^n * sqrt(exp(r)*r*(-1-r+exp(r)*(1+2*r)))), where r is the root of the equation exp(r)*(exp(r) - 1)*r = n.
(a(n)/n!)^(1/n) ~ 2*exp(1/LambertW(2*n)) / LambertW(2*n).
(End)
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/(2^k * k!). - Seiichi Manyama, May 07 2022

A327505 Number of set partitions of [n] where each subset is again partitioned into four nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 65, 350, 1736, 9030, 60355, 561550, 6183221, 69469400, 761767370, 8239194600, 91058524831, 1073790441370, 13900626022985, 196759304278250, 2963381404815566, 46227649788125190, 736940002561065325, 12005645243802471250, 201482801573414254301

Offset: 0

Alois P. Heinz, Sep 14 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..495
Wikipedia, Partition of a set

Column k=4 of A324162.

Cf. A346895.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 4), j=4..n))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j - 1] StirlingS2[j, 4], {j, 4, n}]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/(24^k*k!)); \\ Seiichi Manyama, May 07 2022

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

a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(24^k * k!). - Seiichi Manyama, May 07 2022

A327506 Number of set partitions of [n] where each subset is again partitioned into five nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42651, 253660, 1594230, 12463451, 134921787, 1806386946, 25524454410, 354189159871, 4751404201923, 62042283083648, 803415873180606, 10624141898153091, 148849893975447819, 2279247411153872566, 38395707003954897234

Offset: 0

Alois P. Heinz, Sep 14 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..494
Wikipedia, Partition of a set

Column k=5 of A324162.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 5), j=5..n))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j] Binomial[n - 1, j - 1] StirlingS2[j, 5], {j, 5, n}]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/(120^k*k!)); \\ Seiichi Manyama, May 07 2022

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

a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(120^k * k!). - Seiichi Manyama, May 07 2022

A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! Stirling2(n,k*j)/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 5, 0, 1, 0, 0, 6, 15, 0, 1, 0, 0, 6, 26, 52, 0, 1, 0, 0, 0, 36, 150, 203, 0, 1, 0, 0, 0, 24, 150, 962, 877, 0, 1, 0, 0, 0, 0, 240, 900, 6846, 4140, 0, 1, 0, 0, 0, 0, 120, 1560, 9366, 54266, 21147, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 101556, 471750, 115975, 0

Offset: 0

Seiichi Manyama, Oct 17 2022

			Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  1,   0,   0,   0,   0, ...
  0,  2,   2,   0,   0,   0, ...
  0,  5,   6,   6,   0,   0, ...
  0, 15,  26,  36,  24,   0, ...
  0, 52, 150, 150, 240, 120, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns k=0-4 give: A000007, A000110, A052859, A353664, A353665.

Cf. A324162, A357293, A357868.

T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);

T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));

For k > 0, e.g.f. of column k: exp((exp(x) - 1)^k).

T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * Stirling2(j,k) * T(n-j,k).

A324238 Number of set partitions of [n] where all subsets are partitioned into the same number of nonempty subsets.

Original entry on oeis.org

1, 1, 3, 9, 32, 133, 625, 3328, 20172, 137073, 1023610, 8327069, 73711863, 707141074, 7278630390, 79522233635, 916354807657, 11119419230485, 142082222254701, 1908850117706652, 26862951637197372, 394233330125117457, 6013602782397882264, 95208871146458467659

Offset: 0

Alois P. Heinz, Sep 02 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Partition of a set

Row sums of A324162.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0 or k>n, 0,
      add(b(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
    end:
a:= n-> add(b(n, k), k=0..n):
seq(a(n), n=0..23);

b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0 || k > n, 0, Sum[b[n-j, k]* Binomial[n - 1, j - 1] StirlingS2[j, k], {j, k, n}]]];
a[n_] := Sum[b[n, k], {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, May 05 2020, after Maple *)

A324241 Number of set partitions of [2n] where each subset is again partitioned into n nonempty subsets.

Original entry on oeis.org

1, 2, 10, 100, 1736, 42651, 1324114, 49330996, 2141770488, 106175420065, 5917585057033, 366282501223002, 24930204592110338, 1850568574258750360, 148782988064395367700, 12879868072770703598760, 1194461517469808134322280, 118144018577011379763287565

Offset: 0

Alois P. Heinz, Sep 02 2019

nonn

			a(2) = 10: 123/4, 124/3, 12/34, 134/2, 13/24, 14/23, 1/234, 1/2|3/4, 1/3|2/4, 1/4|2/3.

Alois P. Heinz, Table of n, a(n) for n = 0..345
Wikipedia, Partition of a set

Cf. A007820, A088218, A324162.

b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)
      *binomial(n-1, j-1)*Stirling2(j, k), j=k..n))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..18);

b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0 || k > n, 0, Sum[b[n-j, k]* Binomial[n - 1, j - 1] StirlingS2[j, k], {j, k, n}]]];
a[n_] := b[2n, n];
a /@ Range[0, 18] (* Jean-François Alcover, May 05 2020, after Maple *)

a(n) = if(n==0, 1, stirling(2*n, n, 2)+binomial(2*n, n)/2); \\ Seiichi Manyama, May 08 2022

a(n) = A324162(2n,n).

a(n) = A007820(n) + A088218(n) for n > 0. - Seiichi Manyama, May 08 2022

A325929 Total number of sub-subsets of set partitions of [n] where each subset is again partitioned into nonempty subsets.

Original entry on oeis.org

0, 1, 4, 14, 57, 262, 1326, 7499, 47662, 334794, 2555639, 21124116, 189492474, 1838561337, 19094196270, 210014919406, 2433655645025, 29707254349866, 382324345380310, 5179102279125987, 73515985821539778, 1087888385861343158, 16724494503770495231

Offset: 0

Alois P. Heinz, Sep 08 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A324162.

b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0 or k>n, 0,
      add(b(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
    end:
a:= n-> add(b(n, k)*k, k=0..n):
seq(a(n), n=0..23);

b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 0 || k > n, 0, Sum[b[n - j, k] Binomial[n - 1, j - 1] StirlingS2[j, k], {j, k, n}]]];
a[n_] := Sum[b[n, k] k, {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A324162(n,k).

A327507 Number of set partitions of [n] where each subset is again partitioned into six nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 21, 266, 2646, 22827, 179487, 1324114, 9357348, 64991927, 469882413, 4008715074, 46160063586, 691114045987, 11535301966755, 194240576089826, 3186376950695400, 50592286213334943, 780299037934036929, 11788245937182037114

Offset: 0

Alois P. Heinz, Sep 14 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..494
Wikipedia, Partition of a set

Column k=6 of A324162.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 6), j=6..n))
    end:
seq(a(n), n=0..27);

a(n) = sum(k=0, n\6, (6*k)!*stirling(n, 6*k, 2)/(6!^k*k!)); \\ Seiichi Manyama, May 07 2022

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

a(n) = Sum_{k=0..floor(n/6)} (6*k)! * Stirling2(n,6*k)/(6!^k * k!). - Seiichi Manyama, May 07 2022

A327508 Number of set partitions of [n] where each subset is again partitioned into seven nonempty subsets.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 28, 462, 5880, 63987, 627396, 5715424, 49330996, 408921513, 3292212924, 26136933186, 211891946448, 1910903676319, 21958686224932, 338516695449108, 6257281367040396, 122152192372692405, 2369188918134769500, 44783158458575933110

Offset: 0

Alois P. Heinz, Sep 14 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..496
Wikipedia, Partition of a set

Column k=7 of A324162.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*Stirling2(j, 7), j=7..n))
    end:
seq(a(n), n=0..27);

a(n) = sum(k=0, n\7, (7*k)!*stirling(n, 7*k, 2)/(7!^k*k!)); \\ Seiichi Manyama, May 07 2022

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

a(n) = Sum_{k=0..floor(n/7)} (7*k)! * Stirling2(n,7*k)/(7!^k * k!). - Seiichi Manyama, May 07 2022

cp's OEIS Frontend

A327504 Number of set partitions of [n] where each subset is again partitioned into three nonempty subsets.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A060311 Expansion of e.g.f. exp((exp(x)-1)^2/2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A327505 Number of set partitions of [n] where each subset is again partitioned into four nonempty subsets.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A327506 Number of set partitions of [n] where each subset is again partitioned into five nonempty subsets.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j)/j!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A324238 Number of set partitions of [n] where all subsets are partitioned into the same number of nonempty subsets.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A324241 Number of set partitions of [2n] where each subset is again partitioned into n nonempty subsets.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (kj)! Stirling2(n,k*j)/j!.