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.

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

Views

Author

Ilya Gutkovskiy, Apr 19 2020

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

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*(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

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).

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).

A340822 a(n) = exp(-1) * Sum_{k>=0} (k*(k + n))^n / k!.

Original entry on oeis.org

1, 3, 43, 1211, 54812, 3572775, 313493737, 35368945463, 4962511954307, 844198388785291, 170675800745636572, 40352181663578992883, 11008690527354504977193, 3426969405868832970281647, 1205708016597226199323015459, 475502109963529414669658708847
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2021

Keywords

Crossrefs

Cf. A000110, A020557, A094577, A134980, A334242, A340823.

Programs

  • Mathematica
    Table[Exp[-1] Sum[(k (k + n))^n/k!, {k, 0, Infinity}], {n, 0, 15}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[2 n - k] n^k, {k, 0, n}], {n, 1, 15}]]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(2*n-k) * n^k.

A333761 Decimal expansion of root of the equation LambertW(r) = 1 - r.

Original entry on oeis.org

5, 9, 8, 9, 4, 1, 8, 6, 2, 4, 5, 8, 4, 5, 2, 9, 6, 4, 3, 4, 9, 3, 7, 4, 6, 2, 4, 9, 9, 3, 5, 4, 3, 3, 7, 0, 9, 0, 4, 3, 9, 3, 0, 1, 3, 4, 9, 5, 4, 0, 2, 2, 2, 3, 6, 3, 0, 4, 0, 3, 5, 0, 7, 9, 2, 2, 1, 3, 0, 3, 6, 0, 0, 4, 5, 4, 2, 0, 3, 0, 0, 0, 4, 6, 6, 7, 4, 1, 8, 2, 8, 7, 0, 9, 1, 3, 7, 2, 3, 2, 5, 5, 5, 6, 9, 0
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 09 2020

Keywords

Examples

			0.59894186245845296434937462499354337090439301349540222363040350792213...

Links

Crossrefs

Cf. A030178, A334242.

Programs

  • Mathematica
    RealDigits[r/.FindRoot[LambertW[r] == 1 - r, {r, 1/2}, WorkingPrecision->150], 10, 120][[1]]
  • PARI
    solve(x=0, 1, 1-x-lambertw(x)) \\ Michel Marcus, Jun 09 2020

A337057 a(n) = exp(-n) * Sum_{k>=0} (k - n)^n * n^k / k!.

Original entry on oeis.org

1, 0, 2, 3, 52, 255, 4146, 38766, 688584, 9685017, 195875110, 3655101703, 84872077500, 1955205893680, 51896551499898, 1412668946049315, 42475968202854160, 1328074354724554471, 44778480417250291566, 1577210136570598631318
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2020

Keywords

Crossrefs

Cf. A000296, A194689, A242817, A334242, A335867.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n (Exp[x] - 1 - x)], {x, 0, n}], {n, 0, 19}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] (-n)^(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} binomial(n,k) * (-n)^(n-k) * BellPolynomial_k(n).
Showing 1-6 of 6 results.

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