A334241 -id:A334241 - 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

A335977 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -1, 1, 1, -2, -1, -1, 1, 1, -3, 1, 3, 2, 1, 1, -4, 5, 7, 7, 9, 1, 1, -5, 11, 5, -8, -13, 9, 1, 1, -6, 19, -9, -43, -65, -89, -50, 1, 1, -7, 29, -41, -74, -27, 37, -45, -267, 1, 1, -8, 41, -97, -53, 221, 597, 1024, 1191, -413, 1, 1, -9, 55, -183, 92, 679, 961, 805, 1351, 4723, 2180, 1

Offset: 0

Seiichi Manyama, Jul 03 2020

			Square array begins:
  1,  1,   1,   1,   1,   1,    1, ...
  1,  0,  -1,  -2,  -3,  -4,   -5, ...
  1, -1,  -1,   1,   5,  11,   19, ...
  1, -1,   3,   7,   5,  -9,  -41, ...
  1,  2,   7,  -8, -43, -74,  -53, ...
  1,  9, -13, -65, -27, 221,  679, ...
  1,  9, -89,  37, 597, 961, -341, ...

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

Columns k=0-4 give: A000012, A293037, A309775, A320432, A320433.
Main diagonal gives A334241.
Cf. A292861, A335975.

T[0, k_] := 1; T[n_, k_] := T[n - 1, k] - k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

T(0,k) = 1 and T(n,k) = T(n-1,k) - k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

T(n,k) = exp(k) * Sum_{j>=0} (j + 1)^n * (-k)^j / j!.

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

A335868 a(n) = exp(n) * Sum_{k>=0} (-n)^k * (k - 1)^n / k!.

Original entry on oeis.org

1, -2, 7, -31, 149, -631, 475, 43210, -844727, 10960505, -86569889, -584746911, 46302579229, -1304510879686, 25366896568707, -277053418780891, -4271166460501743, 384590020131637825, -14617527176248527545, 380117694164438489422, -5265650620303861935579

Offset: 0

Ilya Gutkovskiy, Jun 27 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A109747, A292866, A334241, A335867.

Table[n! SeriesCoefficient[Exp[n (1 - Exp[x]) - x], {x, 0, n}], {n, 0, 20}]
Table[Sum[(-1)^(n - k) Binomial[n, k] BellB[k, -n], {k, 0, n}], {n, 0, 20}]

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * BellPolynomial_k(-n).

A334243 a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.

Original entry on oeis.org

1, 0, -2, -3, 44, 245, -2346, -33278, 186808, 6888555, -6774910, -1986368439, -10227075420, 738830661296, 10363304656782, -327255834908715, -9380517430358288, 152180429032236325, 9132761207739810618, -46897839494116200918, -9833058047657527541220

Offset: 0

Ilya Gutkovskiy, Apr 19 2020

sign

Eric Weisstein's World of Mathematics, Bell Polynomial

Cf. A292866, A298373, A334240, A334241, A334242.

Table[n! SeriesCoefficient[Exp[n (1 + x - Exp[x])], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[k, -n] n^(n - k), {k, 0, n}], {n, 1, 20}]]

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

a(n) = Sum_{k=0..n} binomial(n,k) * BellPolynomial_k(-n) * n^(n-k).

A362467 E.g.f. satisfies log(A(x)) = exp(x / A(x)^2) - 1.

Original entry on oeis.org

1, 1, -2, 11, -97, 1162, -17401, 309297, -6284804, 141430821, -3411964025, 84469913200, -1971020234987, 31982106694501, 703663251081166, -131978337454653865, 11571772746664732291, -879307513026396837470, 64266964230133042267891

Offset: 0

Seiichi Manyama, May 26 2023

sign

Cf. A030019, A334241, A349598, A349683, A363302.

Cf. A349654.

a(n) = sum(k=0, n, (-2*n+1)^(k-1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (-2*n+1)^(k-1) * Stirling2(n,k).

A363302 E.g.f. satisfies log(A(x)) = exp(x / A(x)^3) - 1.

Original entry on oeis.org

1, 1, -4, 41, -681, 15667, -460903, 16519141, -698242716, 34004778783, -1874858325725, 115438582354977, -7851013349413919, 584508287058281419, -47281383017104676456, 4129206143361098225405, -387216724567657721607901, 38806186875022459923785751

Offset: 0

Seiichi Manyama, May 26 2023

sign

Cf. A030019, A334241, A349598, A349683, A362467.

Cf. A349655.

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

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

cp's OEIS Frontend

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

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A335977 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(1 - exp(x)) + x).

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

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A335868 a(n) = exp(n) * Sum_{k>=0} (-n)^k * (k - 1)^n / k!.

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A334243 a(n) = exp(n) * Sum_{k>=0} (k + n)^n * (-n)^k / k!.

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A362467 E.g.f. satisfies log(A(x)) = exp(x / A(x)^2) - 1.

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A363302 E.g.f. satisfies log(A(x)) = exp(x / A(x)^3) - 1.

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