A301419 -id:A301419 - cp's OEIS frontend

A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).

1, 1, -1, 1, 25, -674, 15211, -331827, 5987745, 15901597, -13125035449, 1292056076070, -103145930581319, 7462324963409941, -464957409070517453, 16313974895147212801, 2059903411953959582849, -708700955022151333496910, 143215213612865558214820303, -24681846509158429152517973103

Offset: 0

Ilya Gutkovskiy, Aug 20 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..282
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A009235, A014182, A292866, A301419, A317996, A318179, A318180, A318181.

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[(1 - Exp[-n x])/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[(-n)^(n - k) StirlingS2[n, k], {k, n}], {n, 19}]]
Join[{1}, Table[(-n)^n BellB[n, -1/n], {n, 1, 21}]] (* Peter Luschny, Aug 20 2018 *)

{a(n) = sum(k=0, n, (-n)^(n-k)*stirling(n, k, 2))} \\ Seiichi Manyama, Jul 27 2019

a(n) = n! * [x^n] exp((1 - exp(-n*x))/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, Aug 20 2018

A350260 Triangle read by rows. T(n, k) = k^n * BellPolynomial(n, 1/k) for k > 0, if k = 0 then T(n, k) = k^n.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 5, 11, 19, 0, 15, 49, 109, 201, 0, 52, 257, 742, 1657, 3176, 0, 203, 1539, 5815, 15821, 35451, 69823, 0, 877, 10299, 51193, 170389, 447981, 1007407, 2026249, 0, 4140, 75905, 498118, 2032785, 6282416, 16157905, 36458010, 74565473

Offset: 0

Peter Luschny, Dec 22 2021

			Triangle starts:
[0] 1
[1] 0,    1
[2] 0,    2,     3
[3] 0,    5,    11,     19
[4] 0,   15,    49,    109,     201
[5] 0,   52,   257,    742,    1657,    3176
[6] 0,  203,  1539,   5815,   15821,   35451,    69823
[7] 0,  877, 10299,  51193,  170389,  447981,  1007407,  2026249
[8] 0, 4140, 75905, 498118, 2032785, 6282416, 16157905, 36458010, 74565473

Cf. A350256, A350257, A350258, A350259, A350261, A350262, A350263.

Cf. A000110, A301419.

A350260 := (n, k) -> ifelse(k = 0, k^n, k^n * BellB(n, 1/k)):
seq(seq(A350260(n, k), k = 0..n), n = 0..8);

T[n_, k_] := If[k == 0, k^n, k^n BellB[n, 1/k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

A337043 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk - 1)^n / (n^k k!).

Original entry on oeis.org

1, 0, 2, 9, 112, 1875, 43416, 1310946, 49778688, 2313362673, 128894500000, 8469572721533, 647341071298560, 56871349337125648, 5684260661585401728, 640631299771142578125, 80788871646072851660800, 11323828537291632967145015, 1753760620207362607774290432

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

Cf. A000296, A301419, A330605, A334162, A337038, A337039, A337040, A337041, A337042.

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

a(n) = [x^n] (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - n*j*x/(1 + x)).
a(n) = n! * [x^n] exp((exp(n*x) - 1) / n - x), for n > 0.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * n^k * BellPolynomial_k(1/n), for n > 0.

A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk + 1)^n / (n^k k!).

Original entry on oeis.org

1, 2, 6, 35, 352, 5307, 111592, 3117900, 111259904, 4912490375, 261954304224, 16560019685937, 1222893826048000, 104189533522270666, 10132262911996769408, 1114216450970154278543, 137427598621356912082944, 18877351974681584403701519, 2869969478954093766868948480

Offset: 0

Ilya Gutkovskiy, Apr 16 2020

nonn

Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Cf. A000110, A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582, A301419, A334165.

Table[SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^k/Product[(1 - n j x/(1 - x)), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[n! SeriesCoefficient[Exp[x + (Exp[n x] - 1)/n], {x, 0, n}], {n, 1, 18}]]

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=0} (x/(1 - x))^k / Product_{j=1..k} (1 - n*j*x/(1 - x)).
a(n) = n! * [x^n] exp(x + (exp(n*x) - 1) / n), for n > 0.
a(n) = A334165(n,n).

A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

Original entry on oeis.org

1, 1, 4, 33, 456, 9445, 272448, 10386817, 503758720, 30202999821, 2189000524800, 188349613075393, 18954958449853440, 2203304642871358741, 292675996808408743936, 44022321302156791898625, 7438113993194856900034560, 1401876939543892434209075581

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..271

Cf. A000670, A063170, A086914, A094420, A122704, A122778, A229234, A255927, A301419, A326323, A326324.

Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(n - k), {k, 0, n}], {n, 1, 17}]]
Table[SeriesCoefficient[Sum[k! x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n^(n + 1) PolyLog[-n, 1/(n + 1)]/(n + 1), {n, 1, 17}]]

a(n) = sum(k=0, n, stirling(n, k, 2)*k!*n^(n-k)); \\ Michel Marcus, Jan 24 2020

a(n) = [x^n] Sum_{k>=0} k! * x^k / Product_{j=1..k} (1 - n*j*x).
a(n) = n! * [x^n] n / (1 + n - exp(n*x)) for n > 0.
a(n) = n^(n + 1) * Sum_{k>=1} k^n / (n + 1)^(k + 1) for n > 0.
a(n) ~ n! * n^(n+1) / ((n+1) * log(n+1)^(n+1)). - Vaclav Kotesovec, Jun 06 2022

A309401 a(n) = A306245(n,n).

Original entry on oeis.org

1, 1, 3, 43, 5949, 12950796, 586826390263, 669793946192984257, 22558227235537152753501561, 25741074696455818592335996518315259, 1124843928218943684789052411802502269971863691, 2100464404490451025972467064515428575200326254804659324780

Offset: 0

Seiichi Manyama, Jul 28 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..36

Main diagonal of A306245.

Cf. A126443, A301419.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(k^j*binomial(n-1, j)*b(j, k), j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 28 2019

b[0, _] = 1;
b[n_, k_] := b[n, k] = Sum[k^j Binomial[n-1, j] b[j, k], {j, 0, n-1}];
a[n_] := b[n, n];
a /@ Range[0, 12] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)

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 << (0..i - 1).inject(0){|s, j| s + k ** j * ncr(i - 1, j) * ary[j]}}
  ary
end
def A309401(n)
  (0..n).map{|i| A(i, i)}
end
p A309401(20)

cp's OEIS Frontend

A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + n*j*x).

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A350260 Triangle read by rows. T(n, k) = k^n * BellPolynomial(n, 1/k) for k > 0, if k = 0 then T(n, k) = k^n.

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A337043 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (n*k - 1)^n / (n^k * k!).

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A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (n*k + 1)^n / (n^k * k!).

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A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

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A309401 a(n) = A306245(n,n).

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A318183 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 + njx).

A337043 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk - 1)^n / (n^k k!).

A334162 a(0) = 1; thereafter a(n) = exp(-1/n) * Sum_{k>=0} (nk + 1)^n / (n^k k!).