A327504 -id:A327504 - 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

A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).

Original entry on oeis.org

1, 0, 0, 1, 6, 25, 110, 721, 6286, 57625, 541470, 5558641, 64351166, 819480025, 11140978030, 160711583761, 2472834185646, 40597082635225, 706816137889790, 12974021811748081, 250395124862965726, 5074637684604691225, 107798916619788396750

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..452

Cf. A000670, A330047, A346895, A346920.

Cf. A000392, A327504, A346922, A353774.

nmax = 22; CoefficientList[Series[1/(1 - (Exp[x] - 1)^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 06 2021

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(6^k*prod(j=1, 3*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
a(n) ~ n! / (3*(1 + 6^(-1/3)) * log(1 + 6^(1/3))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(6^k * Product_{j=1..3*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/6^k. (End)

A347002 Expansion of e.g.f. exp( -log(1 - x)^3 / 3! ).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 235, 1834, 16352, 163764, 1818030, 22143726, 293476326, 4203311892, 64682865156, 1064154324024, 18636296872320, 346103784493560, 6793394350116600, 140508244952179200, 3054120126193160280, 69596730438090806880, 1659041650323705102840

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..447

Cf. A000399, A327504, A346922, A347001, A347003, A347004.

nmax = 22; CoefficientList[Series[Exp[-Log[1 - x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 3]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/(6^k*k!)); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,3)| * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/(6^k * k!). - Seiichi Manyama, May 06 2022

A353664 Expansion of e.g.f. exp((exp(x) - 1)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 900, 9366, 101556, 1031190, 10995300, 134640726, 1844184276, 26656678230, 400614423300, 6347263038486, 106960986110196, 1905688502565270, 35546025523227300, 691014283378745046, 13999772792477879316, 295570215436360196310

Offset: 0

Seiichi Manyama, May 07 2022

nonn

Cf. A000110, A052859, A353665.

Cf. A327504, A353344, A353774.

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

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(k!*prod(j=1, 3*k, 1-j*x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/k!);

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(k! * Product_{j=1..3*k} (1 - j * x)).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/k!.

A346975 Expansion of e.g.f. log( 1 + (exp(x) - 1)^3 / 3! ).

Original entry on oeis.org

1, 6, 25, 80, 91, -1694, -22875, -193740, -1119569, -768394, 101162425, 1930987240, 23583202371, 181575384906, -306743537075, -45405986594980, -1070132302146089, -16439720013909794, -145808623945689375, 1048196472097011600, 84226169502099763051

Offset: 3

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000392, A327504, A346974, A346976, A346977.

Cf. A346390, A346945.

nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS2[n, 3] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]

a(n) = Stirling2(n,3) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,3) * k * a(k).
a(n) ~ -(n-1)! * 2^(n+1) * cos(n*arctan(2*arctan(3^(5/6)/(2^(2/3) + 3^(1/3))) / log(1 + 6^(1/3) + 6^(2/3)))) / (4*arctan(3^(5/6)/(2^(2/3) + 3^(1/3)))^2 + log(1 + 6^(1/3) + 6^(2/3))^2)^(n/2). - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/3)} (-1)^(k-1) * (3*k)! * Stirling2(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A346390 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^3 / 3! ).

Original entry on oeis.org

1, 6, 25, 100, 511, 3626, 30045, 262800, 2470171, 25889446, 302003065, 3821936300, 51672723831, 745789322466, 11505096936085, 189023074558600, 3288243760145491, 60319276499454686, 1164282909466221105, 23603464830964817700, 501435697062735519151

Offset: 3

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000392, A000629, A003704, A327504, A346894, A346954, A346955.

nmax = 23; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS2[n, 3] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]

my(x='x+O('x^25)); Vec(serlaplace(-log(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 09 2021

a(n) = Stirling2(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,3) * k * a(k).
a(n) ~ (n-1)! / (log(6^(1/3)+1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A353895 Expansion of e.g.f. exp( (x * (exp(x) - 1))^3 / 36 ).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 20, 210, 1400, 7560, 36120, 159390, 850300, 9875580, 170133964, 2688015330, 36706233200, 444802722000, 4939264076016, 52543545234534, 583037908936500, 7645631225897700, 124931080233222340, 2327407301807577066, 44282377224446369800

Offset: 0

Seiichi Manyama, May 09 2022

nonn

Cf. A052506, A353894, A353896.

Cf. A327504, A353884, A353892.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(exp(x)-1))^3/36)))

a(n) = n!*sum(k=0, n\6, (3*k)!*stirling(n-3*k, 3*k, 2)/(36^k*k!*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/6)} (3*k)! * Stirling2(n-3*k,3*k)/(36^k * k! * (n-3*k)!).

A354396 Expansion of e.g.f. exp( -(exp(x) - 1)^3 / 6 ).

Original entry on oeis.org

1, 0, 0, -1, -6, -25, -80, -91, 1694, 23155, 206340, 1442969, 6928394, -6507865, -752409840, -12953182971, -160186016906, -1548849362085, -9789241693220, 28359195353489, 2378650585685794, 52832659521004495, 855581150441210600, 10878338100191146749

Offset: 0

Seiichi Manyama, May 25 2022

sign

Cf. A000587, A354395, A354397, A354398.

Cf. A327504, A354392.

With[{nn=30},CoefficientList[Series[Exp[-(Exp[x]-1)^3/6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 02 2023 *)

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

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, 3, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/((-6)^k*k!));

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,3) * a(n-k).

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

A357032 E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^3 / 6.

Original entry on oeis.org

1, 0, 0, 1, 6, 25, 160, 1981, 24906, 295625, 4044900, 68136541, 1260048086, 24330807865, 508029259920, 11686882860381, 289532464998146, 7588430921962825, 210991834698749020, 6244230552027963901, 195584639712483465486, 6442981074293371848185

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

Cf. A030019, A357031.

Cf. A327504, A357025.

a(n) = sum(k=0, n\3, (3*k)!*(n+1)^(k-1)*stirling(n, 3*k, 2)/(6^k*k!));

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (n+1)^(k-1) * Stirling2(n,3*k)/(6^k * k!).

A354136 Expansion of e.g.f. exp(log(1 + x)^3/6).

Original entry on oeis.org

1, 0, 0, 1, -6, 35, -215, 1414, -9912, 73044, -552570, 4102626, -26654826, 79506492, 2154425364, -73527421176, 1708053626880, -35961691589640, 736338276883080, -15067241745943680, 312009998091705720, -6579362641255341120, 141704946709227843480

Offset: 0

Seiichi Manyama, May 18 2022

sign

Cf. A354137.

Cf. A327504, A347002, A354134.

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

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, 3, 1)*v[i-j+1])); v;

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,3) * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/(6^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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A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).

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A347002 Expansion of e.g.f. exp( -log(1 - x)^3 / 3! ).

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A353664 Expansion of e.g.f. exp((exp(x) - 1)^3).

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A346975 Expansion of e.g.f. log( 1 + (exp(x) - 1)^3 / 3! ).

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Mathematica

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A346390 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^3 / 3! ).

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Mathematica

PARI

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A353895 Expansion of e.g.f. exp( (x * (exp(x) - 1))^3 / 36 ).

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A354396 Expansion of e.g.f. exp( -(exp(x) - 1)^3 / 6 ).

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Mathematica