A334162 -id:A334162 - cp's OEIS frontend

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

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

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.

A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (kj + 1)^n / (k^j j!).

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 6, 15, 1, 2, 7, 24, 52, 1, 2, 8, 35, 116, 203, 1, 2, 9, 48, 214, 648, 877, 1, 2, 10, 63, 352, 1523, 4088, 4140, 1, 2, 11, 80, 536, 3008, 12349, 28640, 21147, 1, 2, 12, 99, 772, 5307, 29440, 112052, 219920, 115975, 1, 2, 13, 120, 1066, 8648, 60389, 324096, 1120849, 1832224, 678570

Offset: 0

Ilya Gutkovskiy, Apr 17 2020

Square array of Dowling numbers.

			Square array begins:
    1,    1,     1,     1,     1,     1,  ...
    2,    2,     2,     2,     2,     2,  ...
    5,    6,     7,     8,     9,    10,  ...
   15,   24,    35,    48,    63,    80,  ...
   52,  116,   214,   352,   536,   772,  ...
  203,  648,  1523,  3008,  5307,  8648,  ...

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

Columns k=1..10 give A000110 (for n > 0), A007405, A003575, A003576, A003577, A003578, A003579, A003580, A003581, A003582.

Cf. A241578, A241579, A334162 (diagonal).

Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[(x/(1 - x))^j/Product[(1 - k i x/(1 - x)), {i, 1, j}], {j, 0, n}], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten
Table[Function[k, n! SeriesCoefficient[Exp[x + (Exp[k x] - 1)/k], {x, 0, n}]][m - n + 1], {m, 0, 10}, {n, 0, m}] // Flatten

G.f. of column k: (1/(1 - x)) * Sum_{j>=0} (x/(1 - x))^j / Product_{i=1..j} (1 - k*i*x/(1 - x)).

E.g.f. of column k: exp(x + (exp(k*x) - 1) / k).

A334193 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, -16, 625, 21384, 571438, 13471744, 188661555, -9794500000, -1476328587789, -134710712340480, -10664210861777200, -744650964057237888, -37832162051689453125, 831929248561267474432, 725944099523076464203157, 167435684777981700601449984

Offset: 0

Ilya Gutkovskiy, Apr 18 2020

sign

Cf. A318183, A334162, A334190, A334191, A334192.

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 + (1 - Exp[n x])/n], {x, 0, n}], {n, 1, 18}]]
Join[{1}, Table[Sum[Binomial[n, k]*n^k*BellB[k, -1/n], {k, 0, n}], {n, 1, 18}]] (* Vaclav Kotesovec, Apr 18 2020 *)

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 + (1 - exp(n*x)) / n), for n > 0.
a(n) = A334192(n,n).

cp's OEIS Frontend

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

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

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A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (kj + 1)^n / (k^j j!).

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A334193 a(0) = 1; thereafter a(n) = exp(1/n) * Sum_{k>=0} (nk + 1)^n / ((-n)^k k!).

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cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

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A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (k*j + 1)^n / (k^j * j!).

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Mathematica

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A334193 a(0) = 1; thereafter a(n) = exp(1/n) * Sum_{k>=0} (n*k + 1)^n / ((-n)^k * k!).

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

A334165 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = exp(-1/k) * Sum_{j>=0} (kj + 1)^n / (k^j j!).

A334193 a(0) = 1; thereafter a(n) = exp(1/n) * Sum_{k>=0} (nk + 1)^n / ((-n)^k k!).