cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A334241 a(n) = exp(n) * Sum_{k>=0} (k + 1)^n * (-n)^k / k!.

Original entry on oeis.org

1, 0, -1, 7, -43, 221, -341, -15980, 370761, -5688125, 62689871, -197586839, -14973562979, 585250669316, -14306382821485, 240985102271971, -1121421968408303, -122020498882279931, 6674724196051810807, -223424819176020519168, 5051515662105879438501
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2020

Keywords

Links

Crossrefs

Cf. A292866, A334193, A334240, A334242, A334243.

Programs

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

Formula

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

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

Views

Author

Ilya Gutkovskiy, Apr 19 2020

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]]

Formula

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

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

Views

Author

Ilya Gutkovskiy, Apr 19 2020

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]]

Formula

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).

A335975 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*(exp(x) - 1) + x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 47, 52, 1, 1, 6, 29, 103, 227, 203, 1, 1, 7, 41, 189, 622, 1215, 877, 1, 1, 8, 55, 311, 1357, 4117, 7107, 4140, 1, 1, 9, 71, 475, 2576, 10589, 29521, 44959, 21147, 1, 1, 10, 89, 687, 4447, 23031, 88909, 227290, 305091, 115975, 1
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2020

Keywords

Examples

			Square array begins:
  1,   1,    1,     1,     1,      1,      1, ...
  1,   2,    3,     4,     5,      6,      7, ...
  1,   5,   11,    19,    29,     41,     55, ...
  1,  15,   47,   103,   189,    311,    475, ...
  1,  52,  227,   622,  1357,   2576,   4447, ...
  1, 203, 1215,  4117, 10589,  23031,  44683, ...
  1, 877, 7107, 29521, 88909, 220341, 478207, ...

Links

Crossrefs

Columns k=0-4 give: A000012, A000110(n+1), A035009(n+1), A078940, A078945.
Main diagonal gives A334240.
Cf. A292860, A335977.

Programs

  • Mathematica
    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, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

Formula

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!.

A335867 a(n) = exp(-n) * Sum_{k>=0} n^k * (k - 1)^n / k!.

Original entry on oeis.org

1, 0, 3, 29, 397, 6879, 144751, 3587100, 102351929, 3305310065, 119186370091, 4746969337923, 206966647324933, 9804683604806908, 501491905963040903, 27544070654283355889, 1616869985889305862385, 101020181695996141703335, 6693303018177050431484035, 468770856837303230888704208
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2020

Keywords

Links

Crossrefs

Cf. A000296, A217924, A242817, A334240, A335868.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n (Exp[x] - 1) - x], {x, 0, n}], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] BellB[k, n], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = n! * [x^n] exp(n*(exp(x) - 1) - x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * BellPolynomial_k(n).

A331345 a(n) = (1/n^2) * Sum_{k>=1} k^n * (1 - 1/n)^(k - 1).

Original entry on oeis.org

1, 3, 37, 1015, 48601, 3583811, 376372333, 53343571695, 9808511445361, 2270198126932219, 645790373135121061, 221449391959470686375, 90084675298978081317961, 42890688646618728144279987, 23627228721958495690763944861, 14910259060767841554203065990111
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 08 2020

Keywords

Links

Crossrefs

Cf. A000670, A050351, A050352, A050353, A094420, A257565, A321189, A334240.

Programs

  • Mathematica
    Join[{1}, Table[1/n^2 Sum[k^n (1 - 1/n)^(k - 1), {k, 1, Infinity}], {n, 2, 16}]]
Table[n! SeriesCoefficient[(Exp[x] - 1)/(Exp[x] - n (Exp[x] - 1)), {x, 0, n}], {n, 1, 16}]

Formula

a(n) = n! * [x^n] (exp(x) - 1) / (exp(x) - n * (exp(x) - 1)).
a(n) = Sum_{k=1..n} Stirling2(n,k) * (n - 1)^(k - 1) * k!.
a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jun 08 2020
Showing 1-6 of 6 results.

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