A242817 -id:A242817 - cp's OEIS frontend

A134954 Number of "hyperforests" on n labeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

1, 1, 2, 8, 55, 562, 7739, 134808, 2846764, 70720278, 2021462055, 65365925308, 2359387012261, 94042995460130, 4102781803365418, 194459091322828280, 9950303194613104995, 546698973373090998382, 32101070021048906407183, 2006125858248695722280564

Offset: 0

Don Knuth, Jan 26 2008

nonn

The part of the name "assuming that each edge contains at least two vertices" is ambiguous. It may mean that not all n vertices have to be covered by some edge of the hypergraph, i.e., it is not necessarily a spanning hyperforest. However it is common to represent uncovered vertices as singleton edges, as in my example. - Gus Wiseman, May 20 2018

			From _Gus Wiseman_, May 20 2018: (Start)
The a(3) = 8 labeled spanning hyperforests are the following:
{{1,2,3}}
{{1,3},{2,3}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}
(End)

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H. - Washington Bomfim, Sep 25 2008

Alois P. Heinz, Table of n, a(n) for n = 0..370
N. J. A. Sloane, Transforms

Cf. A030019, A035053, A048143, A054921, A134955, A134956, A134957, A144959, A242817, A304716, A304717, A304867, A304911, A304912.

b:= proc(n) option remember; add(Stirling2(n-1,i) *n^(i-1), i=0..n-1) end: B:= proc(n) x-> add(b(k) *x^k/k!, k=0..n) end: a:= n-> coeff(series(exp(B(n)(x)), x, n+1), x,n) *n!: seq(a(n), n=0..30);  # Alois P. Heinz, Sep 09 2008

b[n_] := b[n] = Sum[StirlingS2[n-1, i]*n^(i-1), {i, 0, n-1}]; B[n_][x_] := Sum[b[k] *x^k/k!, {k, 0, n}]; a[0]=1; a[n_] := SeriesCoefficient[ Exp[B[n][x]], {x, 0, n}] *n!; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Exponential transform of A030019. - N. J. A. Sloane, Jan 30 2008
Binomial transform of A304911. - Gus Wiseman, May 20 2018
a(n) = Sum of n!*Product_{k=1..n} (A030019(k)/k!)^c_k / (c_k)! over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0. - Washington Bomfim, Sep 25 2008
a(n) ~ exp((n+1)/LambertW(1)) * n^(n-2) / (sqrt(1+LambertW(1)) * exp(2*n+2) * (LambertW(1))^n). - Vaclav Kotesovec, Jul 26 2014

A189233 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals upwards, where the e.g.f. of column k is exp(k*(e^x-1)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 94, 57, 20, 5, 1, 0, 203, 454, 309, 116, 30, 6, 1, 0, 877, 2430, 1866, 756, 205, 42, 7, 1, 0, 4140, 14214, 12351, 5428, 1555, 330, 56, 8, 1, 0, 21147, 89918, 88563, 42356, 12880, 2850, 497, 72, 9, 1

Offset: 0

Peter Luschny, Apr 18 2011

A(n,k) is the n-th moment of a Poisson distribution with mean = k. - Geoffrey Critzer, Dec 23 2018

			Square array begins:
       A000007 A000110 A001861 A027710 A078944 A144180 A144223 A144263
A000012   1,    1,    1,    1,    1,     1,     1,     1, ...
A001477   0,    1,    2,    3,    4,     5,     6,     7, ...
A002378   0,    2,    6,   12,   20,    30,    42,    56, ...
A033445   0,    5,   22,   57,  116,   205,   330,   497, ...
          0,   15,   94,  309,  756,  1555,  2850,  4809, ...
          0,   52,  454, 1866, 5428, 12880, 26682, 50134, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5150
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Peter Luschny, Set partitions and Bell numbers

Columns: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263.
Rows: A000012, A001477, A002378, A033445.
Main diagonal gives A242817.
Cf. A000587, A144150.

# Cf. also the Maple prog. of Alois P. Heinz in A144223 and A144180.
expnums := proc(k,n) option remember; local j;
`if`(n = 0, 1, (1+add(binomial(n-1,j-1)*expnums(k,n-j), j = 1..n-1))*k) end:
A189233_array := (k, n) -> expnums(k,n):
seq(print(seq(A189233_array(k,n), k = 0..7)), n = 0..5);
A189233_egf := k -> exp(k*(exp(x)-1));
T := (n,k) -> n!*coeff(series(A189233_egf(k), x, n+1), x, n):
seq(lprint(seq(T(n,k), k = 0..7)), n = 0..5):
# alternative Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

max = 9; Clear[col]; col[k_] := col[k] = CoefficientList[ Series[ Exp[k*(Exp[x]-1)], {x, 0, max}], x]*Range[0, max]!; a[0, ] = 1; a[n?Positive, 0] = 0; a[n_, k_] := col[k][[n+1]]; Table[ a[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)
Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid  (* Geoffrey Critzer, Dec 23 2018 *)

A(n,k):=if k=0 and n=0 then 1 else if k=0 then 0 else  sum(stirling2(n,i)*k^i,i,0,n); /* Vladimir Kruchinin, Apr 12 2019 */

E.g.f. of column k: exp(k*(e^x-1)).
A(n,1) = A000110(n), A(n, -1) = A000587(n).
A(n,k) = BellPolynomial(n, k). - Geoffrey Critzer, Dec 23 2018
A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - Vladimir Kruchinin, Apr 12 2019

A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 0

Offset: 0

Seiichi Manyama, Sep 25 2017

			Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,   1,    2,     3,     4,      5,      6, ...
   0,   2,    6,    12,    20,     30,     42, ...
   0,   5,   22,    57,   116,    205,    330, ...
   0,  15,   94,   309,   756,   1555,   2850, ...
   0,  52,  454,  1866,  5428,  12880,  26682, ...
   0, 203, 2430, 12351, 42356, 115155, 268098, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.
Rows n=0..2 give A000012, A001477, A002378.
Main diagonal gives A242817.
Same array, different indexing is A189233.
Cf. A292861.

A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[, ] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, k). - Peter Luschny, Dec 23 2021

A292866 a(n) = n! * [x^n] exp(n*(1 - exp(x))).

Original entry on oeis.org

1, -1, 2, -3, -20, 370, -4074, 34293, -138312, -2932533, 106271090, -2192834490, 32208497124, -206343936097, -7657279887698, 412496622532785, -12455477719752976, 260294034150380430, -2256541295745391542, -122593550603339550843, 8728842979656718306780

Offset: 0

Seiichi Manyama, Sep 25 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..415

Main diagonal of A292861.

Cf. A242817.

b:= proc(n, k) option remember; `if`(n=0, 1,
      -(1+add(binomial(n-1, j-1)*b(n-j, k), j=1..n-1))*k)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 25 2017

Table[n!*SeriesCoefficient[E^(n*(1 - E^x)),{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 25 2017 *)
a[n_] := BellB[n, -n]; Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 23 2021 *)

{a(n) = sum(k=0, n, (-n)^k*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 28 2019

def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << k * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[j]}}
  ary
end
def A292866(n)
  (0..n).map{|i| A(-i, i)[-1]}
end
p A292866(20)

a(n) = exp(n) * Sum_{k>=0} (-n)^k*k^n/k!. - Ilya Gutkovskiy, Jul 13 2019
a(n) = Sum_{k=0..n} (-n)^k * Stirling2(n,k). - Seiichi Manyama, Jul 28 2019
a(n) = BellPolynomial(n, -n). - Peter Luschny, Dec 23 2021

A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - njx).

Original entry on oeis.org

1, 1, 3, 19, 201, 3176, 69823, 2026249, 74565473, 3376695763, 183991725451, 11854772145800, 890415496931689, 77023751991841669, 7592990698770559111, 845240026276785888451, 105409073489605774592897, 14625467507717709778793020, 2244123413703647502288608467, 378751257186051653931253015229

Offset: 0

Ilya Gutkovskiy, Mar 20 2018

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..101
N. J. A. Sloane, Transforms

Cf. A000110, A004211, A004212, A004213, A005011, A005012, A008277, A075506, A075507, A075508, A075509, A242817, A292914, A318183.

List([0..20],n->Sum([0..n],k->n^(n-k)*Stirling2(n,k))); # Muniru A Asiru, Mar 20 2018

Table[SeriesCoefficient[Sum[x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[n^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 19}]]
(* Or: *)
A301419[n_] := If[n == 0, 1, n^n BellB[n, 1/n]];
Table[A301419[n], {n, 0, 19}] (* Peter Luschny, Dec 22 2021 *)

a(n) = sum(k=0, n, n^(n-k)*stirling(n, k, 2)); \\ Michel Marcus, Mar 23 2018

a(n) = n! * [x^n] exp((exp(n*x) - 1)/n), for n > 0.
a(n) = Sum_{k=0..n} n^(n-k)*Stirling2(n,k).
a(n) = n^n * BellPolynomial(n, 1/n) for n >= 1. - Peter Luschny, Dec 22 2021
a(n) ~ exp(n/LambertW(n^2) - n) * n^(2*n) / (sqrt(1 + LambertW(n^2)) * LambertW(n^2)^n). - Vaclav Kotesovec, Jun 06 2022

A317172 a(n) = n! * [x^n] 1/(1 - n*log(1 + x)).

Original entry on oeis.org

1, 1, 6, 114, 4168, 248870, 21966768, 2685571560, 434202400896, 89679267601632, 23032451508686400, 7199033431349412576, 2690461258552995849216, 1184680716090974803461072, 606986901206377433194091520, 358023049940533240478842992000, 240858598980174362552808566194176

Offset: 0

Ilya Gutkovskiy, Jul 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A006252, A088501, A094420, A242817, A317171.

Main diagonal of A320080.

Table[n! SeriesCoefficient[1/(1 - n Log[1 + x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[StirlingS1[n, k] n^k k!, {k, n}], {n, 16}]]

{a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 1))} \\ Seiichi Manyama, Jun 12 2020

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

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jul 23 2018

A334240 a(n) = exp(-n) * Sum_{k>=0} (k + 1)^n * n^k / k!.

Original entry on oeis.org

1, 2, 11, 103, 1357, 23031, 478207, 11741094, 332734521, 10689163687, 383851610331, 15236978883127, 662491755803269, 31311446539427926, 1598351161031967063, 87638233726766111731, 5136809177699534717169, 320521818480481139673919, 21212211430440994022892019

Offset: 0

Ilya Gutkovskiy, Apr 19 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..368
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A242817, A299824, A334162, A334241, A334242, A334243.

Table[n! SeriesCoefficient[Exp[x + n (Exp[x] - 1)], {x, 0, n}], {n, 0, 18}]
Table[Sum[Binomial[n, k] BellB[k, n], {k, 0, n}], {n, 0, 18}]

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (n*x/(1 - x))^k / Product_{j=1..k} (1 - j*x/(1 - x)).
a(n) = n! * [x^n] exp(x + n*(exp(x) - 1)).
a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(n).
a(n) ~ exp((1/LambertW(1) - 2)*n) * n^n / (sqrt(1+LambertW(1)) * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jun 08 2020

A350256 Triangle read by rows. T(n, k) = BellPolynomial(n, k).

Original entry on oeis.org

1, 0, 1, 0, 2, 6, 0, 5, 22, 57, 0, 15, 94, 309, 756, 0, 52, 454, 1866, 5428, 12880, 0, 203, 2430, 12351, 42356, 115155, 268098, 0, 877, 14214, 88563, 355636, 1101705, 2869242, 6593839, 0, 4140, 89918, 681870, 3188340, 11202680, 32510850, 82187658, 187104200

Offset: 0

Peter Luschny, Dec 22 2021

			Triangle begins:
[0] 1
[1] 0,    1
[2] 0,    2,     6
[3] 0,    5,    22,     57
[4] 0,   15,    94,    309,     756
[5] 0,   52,   454,   1866,    5428,    12880
[6] 0,  203,  2430,  12351,   42356,   115155,   268098
[7] 0,  877, 14214,  88563,  355636,  1101705,  2869242,  6593839
[8] 0, 4140, 89918, 681870, 3188340, 11202680, 32510850, 82187658, 187104200

Cf. A242817 (main diagonal), A000110 (column 1), A350264 (row sums), A350263 (Bell(n,-k)), A189233 and A292860 (array).

A350256 := (n, k) -> ifelse(n = 0, 1, BellB(n, k)):
seq(seq(A350256(n, k), k = 0..n), n = 0..8);

T[n_, k_] := BellB[n, k]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

A334242 a(n) = exp(-n) * Sum_{k>=0} (k + n)^n * n^k / k!.

Original entry on oeis.org

1, 2, 18, 273, 5812, 159255, 5336322, 211385076, 9663571400, 500742188415, 29002424377110, 1856728690107027, 130194428384173116, 9923500366931329282, 816909605562423271178, 72231668379957026776065, 6827368666949651984215824, 686970682778467688690704639

Offset: 0

Ilya Gutkovskiy, Apr 19 2020

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A134980, A242817, A334240, A334241, A334243.

Table[n! SeriesCoefficient[Exp[n (Exp[x] + x - 1)], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[k, n] n^(n - k), {k, 0, n}], {n, 1, 17}]]

a(n) = n! * [x^n] exp(n*(exp(x) + x - 1)).
a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(n) * n^(n-k).
a(n) ~ c * exp((r^2/(1-r) - 1)*n) * n^n / (1-r)^n, where r = A333761 = 0.59894186245845296434937... is the root of the equation LambertW(r) = 1-r and c = 0.897950293373062982395233981707095204244165706668136925178217032608352851... - Vaclav Kotesovec, Jun 09 2020

A299824 a(n) = (1/e^n)Sum_{j >= 1} j^n n^j / (j-1)!.

Original entry on oeis.org

2, 22, 309, 5428, 115155, 2869242, 82187658, 2661876168, 96202473183, 3838516103310, 167606767714397, 7949901069639228, 407048805012563038, 22376916254447538882, 1314573505901491675965, 82188946843192555474704, 5448870914168179374456623, 381819805747937892412056342

Offset: 1

Pedro Caceres, Feb 19 2018

nonn

For m>1, A242817(m) and a(m-1) are also the m-th and (m+1)-st terms of the sequences "Number of ways of placing X labeled balls into X unlabeled (but (m-1)-colored) boxes". For instance, sequence A144180 for 5-colored boxes (m = 6), has A144180(6) = 12880, and A144180(7) = 115155, which are A242817(6) and a(5) respectively. Same pattern can be observed for A027710, A144223, A144263 (comment added after Omar E. Pol's formula).

			a(4) = (1/e^4)*Sum_{j >= 1} j^4 * 4^j / (j-1)! = 5428.

Alois P. Heinz, Table of n, a(n) for n = 1..368

Cf. A027710, A144180, A144223, A144263, A189233, A242817, A292860.

a(n) = round(exp(-n)*suminf(j = 1, (j^n)*(n^j)/(j-1)!)); \\ Michel Marcus, Feb 24 2018

A299824(n,f=exp(n),S=n/f,t)=for(j=2,oo,S+=(t=j^n*n^j)/(f*=j-1);tn&&return(ceil(S))) \\ For n > 23, use \p## with some ## >= 2n. - M. F. Hasler, Mar 09 2018

a(n) = A189233(n+1,n). - Omar E. Pol, Feb 24 2018
a(n) ~ exp(n/LambertW(1) - 2*n) * n^(n + 1) / (sqrt(1 + LambertW(1)) * LambertW(1)^(n + 1)). - Vaclav Kotesovec, Mar 08 2018
Or: a(n) ~ (1/sqrt(1+w)) * exp(1/w-2)^n * (n/w)^(n+1), with w = LambertW(1) ~ 0.56714329... The relative error decreases from 10^-2 for a(2) to 10^-3 for a(15), but reaches 10^-3.5 only at a(45). - M. F. Hasler, Mar 09 2018

cp's OEIS Frontend

A134954 Number of "hyperforests" on n labeled nodes, i.e., hypergraphs that have no cycles, assuming that each edge contains at least two vertices.

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A189233 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals upwards, where the e.g.f. of column k is exp(k*(e^x-1)).

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A292860 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1)).

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A292866 a(n) = n! * [x^n] exp(n*(1 - exp(x))).

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A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - n*j*x).

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A317172 a(n) = n! * [x^n] 1/(1 - n*log(1 + x)).

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A334240 a(n) = exp(-n) * Sum_{k>=0} (k + 1)^n * n^k / k!.

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A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - njx).

A299824 a(n) = (1/e^n)Sum_{j >= 1} j^n n^j / (j-1)!.