A327506 -id:A327506 - cp's OEIS frontend

A324162 Number T(n,k) of set partitions of [n] where each subset is again partitioned into k nonempty subsets; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 5, 3, 1, 0, 15, 10, 6, 1, 0, 52, 45, 25, 10, 1, 0, 203, 241, 100, 65, 15, 1, 0, 877, 1428, 511, 350, 140, 21, 1, 0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1, 0, 21147, 67035, 29765, 9030, 6951, 2646, 462, 36, 1, 0, 115975, 524926, 250200, 60355, 42651, 22827, 5880, 750, 45, 1

Offset: 0

Alois P. Heinz, Sep 02 2019

			T(4,2) = 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.
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    2,    1;
  0,    5,    3,    1;
  0,   15,   10,    6,    1;
  0,   52,   45,   25,   10,    1;
  0,  203,  241,  100,   65,   15,   1;
  0,  877, 1428,  511,  350,  140,  21,  1;
  0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A000110 (for n>0), A060311, A327504, A327505, A327506, A327507, A327508, A327509, A327510, A327511.
Row sums give A324238.
T(2n,n) gives A324241.
Cf. A008277, A048993, A039810, A130191, A325929.

T:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0, add(
      T(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);

nmax = 10;
col[k_] := col[k] = CoefficientList[Exp[(Exp[x]-1)^k/k!] + O[x]^(nmax+1), x][[k+1;;]] Range[k, nmax]!;
T[n_, k_] := Which[k == n, 1, k == 0, 0, True, col[k][[n-k+1]]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 26 2020 *)

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

E.g.f. of column k>0: exp((exp(x)-1)^k/k!).
Sum_{k=1..n} k * T(n,k) = A325929(n).
T(n,k) = Sum_{j=0..floor(n/k)} (k*j)! * Stirling2(n,k*j)/(k!^j * j!) for k > 0. - Seiichi Manyama, May 07 2022

A347004 Expansion of e.g.f. exp( -log(1 - x)^5 / 5! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269451, 3423860, 46238280, 664233856, 10143487354, 164423078456, 2823768543960, 51272283444264, 982177492263750, 19807082824819374, 419629806223448346, 9320808413229618816, 216645165604679499072, 5259724543984442886486

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..448

Cf. A000482, A327506, A346924, A347001, A347002, A347003.

nmax = 23; CoefficientList[Series[Exp[-Log[1 - x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/(120^k*k!)); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,5)| * a(n-k).

a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/(120^k * k!). - Seiichi Manyama, May 06 2022

A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42777, 260590, 1809060, 17418401, 229768539, 3402511476, 50013258750, 706670789371, 9659104177101, 130958047050698, 1834295186003784, 27849428308615221, 472297857494304303, 8856291348143365456, 176841068643273207426

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..469

Cf. A000670, A330047, A346894, A346895.

Cf. A000481, A327506, A346924.

nmax = 24; CoefficientList[Series[1/(1 - (Exp[x] - 1)^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^5/5!))) \\ Michel Marcus, Aug 07 2021

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (5*k)!*x^(5*k)/(120^k*prod(j=1, 5*k, 1-j*x)))) \\ Seiichi Manyama, May 09 2022

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/120^k); \\ Seiichi Manyama, May 09 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) ~ n! / (5*(1 + 120^(-1/5)) * log(1 + 120^(1/5))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 09 2022: (Start)
G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(120^k * Product_{j=1..5*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/120^k. (End)

A346977 Expansion of e.g.f. log( 1 + (exp(x) - 1)^5 / 5! ).

Original entry on oeis.org

1, 15, 140, 1050, 6951, 42399, 239800, 1164570, 2553551, -54771717, -1384474728, -23286667950, -339924740609, -4554547609233, -56481301888144, -630768487283886, -5665064764515849, -18095553874845909, 924820173031946752, 35413415495503624986

Offset: 5

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000481, A327506, A346955, A346974, A346975, A346976.

Cf. A346947, A346955.

nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

a(n) = Stirling2(n,5) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ -(n-1)! * 2^(1+n) * 5^n * cos(n*arctan((2*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))) / log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))) / (100*arctan(sqrt(10 - 2*sqrt(5))/(1 + sqrt(5) + 2^(7/5)/15^(1/5)))^2 + (5*log(1 + 3^(1/5)*5^(7/10)/2^(2/5) + 15^(1/5)/2^(2/5) + 2^(6/5)*15^(2/5)))^2)^(n/2). - Vaclav Kotesovec, Aug 10 2021
a(n) = Sum_{k=1..floor(n/5)} (-1)^(k-1) * (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A354398 Expansion of e.g.f. exp( -(exp(x) - 1)^5 / 120 ).

Original entry on oeis.org

1, 0, 0, 0, 0, -1, -15, -140, -1050, -6951, -42399, -239800, -1164570, -2553551, 54771717, 1384600854, 23301803070, 340911045929, 4600861076433, 58236569430172, 687816515641206, 7315220762286129, 61629305427537309, 140107851269900954, -11001310744922517426

Offset: 0

Seiichi Manyama, May 25 2022

sign

Cf. A000587, A354395, A354396, A354397.

Cf. A327506, A346977, A354394.

With[{nn=30},CoefficientList[Series[Exp[-(Exp[x]-1)^5/120],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 23 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^5/120)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/((-120)^k*k!));

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,5) * a(n-k).

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

A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

Original entry on oeis.org

1, 15, 140, 1050, 6951, 42651, 253660, 1594230, 12463451, 134921787, 1806513072, 25539589530, 355175465191, 4797717669123, 63797550625676, 860468790181686, 12275324511112971, 192498455326842819, 3353266112959628272, 63379650000684213834

Offset: 5

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000481, A000629, A003704, A327506, A346390, A346920, A346954.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS2[n, 5] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 24}]

a(n) = Stirling2(n,5) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,5) * k * a(k).
a(n) ~ (n-1)! / (log(120^(1/5) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/5)} (5*k)! * Stirling2(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A354137 Expansion of e.g.f. exp(log(1 + x)^5/120).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, -15, 175, -1960, 22449, -269199, 3410000, -45753180, 650179816, -9771920158, 155020385156, -2589888417480, 45461879164584, -836540418765834, 16099972965770778, -323385447259166454, 6764948641797695496, -147088325599708573080

Offset: 0

Seiichi Manyama, May 18 2022

sign

Cf. A354136.

Cf. A327506, A347004, A354135.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^5/120)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 1)*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1)/(120^k*k!));

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,5) * a(n-k).

a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling1(n,5*k)/(120^k * k!).

cp's OEIS Frontend

A324162 Number T(n,k) of set partitions of [n] where each subset is again partitioned into k nonempty subsets; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A347004 Expansion of e.g.f. exp( -log(1 - x)^5 / 5! ).

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A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

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A346977 Expansion of e.g.f. log( 1 + (exp(x) - 1)^5 / 5! ).

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A354398 Expansion of e.g.f. exp( -(exp(x) - 1)^5 / 120 ).

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A346955 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^5 / 5! ).

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A354137 Expansion of e.g.f. exp(log(1 + x)^5/120).

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