A327505 -id:A327505 - 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

A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 65, 350, 1771, 10290, 86605, 977350, 11778041, 138208070, 1590920695, 18895490250, 245692484311, 3587464083850, 57397496312585, 966066470023550, 16713560617838581, 297182550111615630, 5500448659383161275, 107267326981597659250

Offset: 0

Ilya Gutkovskiy, Aug 06 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..461

Cf. A000670, A330047, A346894, A346920.

Cf. A000453, A327505, A346923, A353775.

nmax = 23; CoefficientList[Series[1/(1 - (Exp[x] - 1)^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^4/4!))) \\ Michel Marcus, Aug 06 2021

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(24^k*prod(j=1, 4*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, 4, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022

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

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
a(n) ~ n! / (4*(1 + 2^(-3/4)*3^(-1/4)) * log(1 + 2^(3/4)*3^(1/4))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(24^k * Product_{j=1..4*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/24^k. (End)

A347003 Expansion of e.g.f. exp( log(1 - x)^4 / 4! ).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6804, 68544, 754130, 9044750, 117773656, 1656897528, 25061576176, 405667844400, 6997383182854, 128126051451184, 2481884332498848, 50702417505257904, 1089371806098805286, 24555007848629510700, 579348221233739760550, 14278529041496660104450

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

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

Cf. A000454, A327505, A346923, A347001, A347002, A347004.

nmax = 23; CoefficientList[Series[Exp[Log[1 - x]^4/4!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 4]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/(24^k*k!)); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,4)| * a(n-k).

a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/(24^k * k!). - Seiichi Manyama, May 06 2022

A353665 Expansion of e.g.f. exp((exp(x) - 1)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 1560, 8400, 60984, 912240, 15938520, 242998800, 3300493944, 44583979440, 690641504280, 12868117189200, 264164524958904, 5481631005177840, 112822632387018840, 2367468210865875600, 52624238539033647864, 1258531092544541563440

Offset: 0

Seiichi Manyama, May 07 2022

nonn

Cf. A000110, A052859, A353664.

Cf. A327505, A353358, A353775.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^4)))

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(k!*prod(j=1, 4*k, 1-j*x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 4, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/k!);

E.g.f.: exp((exp(x) - 1)^4).
G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(k! * Product_{j=1..4*k} (1 - j * x)).
a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,4) * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/k!.

A346976 Expansion of e.g.f. log( 1 + (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1666, 6510, 7855, -270050, -4942894, -63052990, -682650605, -6309889950, -42960995804, 348211510, 7739540496935, 202902567668150, 3863986259609686, 61527382177040010, 807717870749781475, 7066953051021894250, -33781117662453993424

Offset: 4

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000453, A327505, A346954, A346974, A346975, A346977.

Cf. A346946, A346954.

nmax = 24; CoefficientList[Series[Log[1 + (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

a(n) = Stirling2(n,4) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ -(n-1)! * 2^(n+1) * cos(n*arctan((2*arctan(1/(1 + 1/6^(1/4)))) / log(1 + 2*6^(1/4) + 2*6^(1/2)))) / (4*arctan(1/(1 + 1/6^(1/4)))^2 + log(1 + 2*6^(1/4) + 2*6^(1/2))^2)^(n/2). - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (-1)^(k-1) * (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A354397 Expansion of e.g.f. exp( -(exp(x) - 1)^4 / 24 ).

Original entry on oeis.org

1, 0, 0, 0, -1, -10, -65, -350, -1666, -6510, -7855, 270050, 4948669, 63503440, 702095030, 6924754200, 58870214129, 356043924590, -615569993285, -74306502570650, -1783956267419536, -32695418069393310, -520090808927130925, -7317310078355307250, -87056749651694635451

Offset: 0

Seiichi Manyama, May 25 2022

sign

Cf. A000587, A354395, A354396, A354398.

Cf. A327505, A354393.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^4/24)))

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

a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/((-24)^k*k!));

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,4) * a(n-k).

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

A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

Original entry on oeis.org

1, 10, 65, 350, 1736, 9030, 60355, 561550, 6188996, 69919850, 781211795, 8854058850, 106994019406, 1433756147470, 21287253921635, 339206526695750, 5630710652048216, 96341917117951890, 1708973354556320875, 31787279786739738250, 623964823224788294426

Offset: 4

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000453, A000629, A003704, A327505, A346390, A346895, A346955.

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

a(n) = Stirling2(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).
a(n) ~ (n-1)! / (log(2^(3/4)*3^(1/4) + 1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A353896 Expansion of e.g.f. exp( (x * (exp(x) - 1))^4 / 576 ).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 70, 1260, 13650, 115500, 841995, 5555550, 34139105, 198948750, 1144463320, 8171563400, 112204064700, 2364061354200, 49912312090845, 951208121086650, 16403948060775275, 260328078068154250, 3860274855288458376, 54182965918066177500

Offset: 0

Seiichi Manyama, May 09 2022

nonn

Cf. A052506, A353894, A353895.

Cf. A327505, A353885, A353893.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(exp(x)-1))^4/576)))

a(n) = n!*sum(k=0, n\8, (4*k)!*stirling(n-4*k, 4*k, 2)/(576^k*k!*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/8)} (4*k)! * Stirling2(n-4*k,4*k)/(576^k * k! * (n-4*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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A346895 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^4 / 4!).

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A347003 Expansion of e.g.f. exp( log(1 - x)^4 / 4! ).

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A353665 Expansion of e.g.f. exp((exp(x) - 1)^4).

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A346976 Expansion of e.g.f. log( 1 + (exp(x) - 1)^4 / 4! ).

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A354397 Expansion of e.g.f. exp( -(exp(x) - 1)^4 / 24 ).

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A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

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Mathematica

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A353896 Expansion of e.g.f. exp( (x * (exp(x) - 1))^4 / 576 ).

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