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.

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A308645 Expansion of e.g.f. exp(1 + x - exp(2*x)).

Original entry on oeis.org

1, -1, -3, 3, 41, 87, -571, -5701, -14575, 156655, 2094925, 9148851, -63364423, -1474212665, -11494853995, 10945362411, 1520718442785, 20719421344991, 100137575499165, -1638818071763869, -45333849658449847, -512404024891840969, -577060092568365467, 99142586163648127771
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 13 2019

Keywords

Crossrefs

Cf. A000587, A055882, A126390, A308536.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[1 + x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Exp[1] Sum[(-1)^k (2 k + 1)^n/k!, {k, 0, Infinity}], {n, 0, 23}]
Table[Sum[Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 23}]

Formula

a(n) = exp(1) * Sum_{k>=0} (-1)^k*(2*k + 1)^n/k!.
a(n) = Sum_{k=0..n} binomial(n,k)*2^k*A000587(k).

A355162 a(n) = exp(-1) * Sum_{k>=0} (4*k + 2)^n / k!.

Original entry on oeis.org

1, 6, 52, 568, 7312, 107360, 1760576, 31760256, 623137024, 13179872768, 298391335936, 7189153167360, 183428957442048, 4935794590572544, 139571328018628608, 4134634425826115584, 127966201403431518208, 4127825849826169716736, 138477447400991610896384, 4822002684952714247929856
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 22 2022

Keywords

Crossrefs

Cf. A000110, A005493, A126390, A284859, A284864, A285064, A355163.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 2 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] 2^(n + k) BellB[k], {k, 0, n}], {n, 0, 19}]

Formula

E.g.f.: exp(exp(4*x) + 2 x - 1).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n+k) * Bell(k).
a(n) = 2^n * A126390(n). - Vaclav Kotesovec, Jun 22 2022
a(n) ~ 4^n * n^(n + 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 1/2)). - Vaclav Kotesovec, Jun 27 2022

A355163 a(n) = exp(-1) * Sum_{k>=0} (4*k + 3)^n / k!.

Original entry on oeis.org

1, 7, 65, 743, 9921, 150151, 2526593, 46615783, 933072513, 20093861895, 462440842177, 11310514854375, 292627518129985, 7976748158144647, 228308400790500097, 6840702405678586343, 214000748166439723265, 6973447420429351808007, 236204029044752265931585, 8300724166287243795922151
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 22 2022

Keywords

Crossrefs

Cf. A000110, A005494, A126390, A284859, A284864, A285064, A355162.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Exp[4 x] + 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 3 a[n - 1] + Sum[Binomial[n - 1, k - 1] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
Table[Sum[Binomial[n, k] 3^(n - k) 4^k BellB[k], {k, 0, n}], {n, 0, 19}]

Formula

E.g.f.: exp(exp(4*x) + 3 x - 1).
a(0) = 1; a(n) = 3 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 4^k * Bell(k).
a(n) ~ Bell(n) * (4 + 3*LambertW(n)/n)^n. - Vaclav Kotesovec, Jun 22 2022
a(n) ~ 4^n * n^(n + 3/4) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n + 3/4)). - Vaclav Kotesovec, Jun 27 2022

A367743 Expansion of e.g.f. exp(1 - x - exp(2*x)).

Original entry on oeis.org

1, -3, 5, 1, -7, -75, -99, 1241, 10161, 18989, -332299, -3857551, -14440151, 141168997, 2807256333, 20182451657, -42073176479, -2999363709091, -38439478980891, -161835672017439, 3439471815545177, 87228227501354517, 937579822282327421, 216540362854403513, -198501712690150659055
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 29 2023

Keywords

Crossrefs

Cf. A000587, A004211, A009235, A109747, A124311, A126390, A308536, A308645, A367744.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[1 - x - Exp[2 x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] 2^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, -1], {k, 0, n}], {n, 0, 24}]

Formula

a(n) = exp(1) * Sum_{k>=0} (-1)^k * (2*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 2^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * A000587(k).

A367946 Expansion of e.g.f. exp(2*(exp(2*x) - 1) + x).

Original entry on oeis.org

1, 5, 33, 261, 2369, 24069, 269153, 3272453, 42858113, 600181765, 8933677729, 140645797125, 2332169258945, 40586333768197, 738998405168609, 14040304543111941, 277678389593341185, 5704502830382733317, 121500343635119818017, 2678407616841000605957, 61015572313688043492929
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 05 2023

Keywords

Crossrefs

Cf. A000110, A035009, A126390, A308543, A367945.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[Exp[2 (Exp[2 x] - 1) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(2*x) - 1) + x))) \\ Michel Marcus, Dec 07 2023

Formula

G.f. A(x) satisfies: A(x) = 1 + x * ( A(x) + 4 * A(x/(1 - 2*x)) / (1 - 2*x) ).
a(n) = exp(-2) * Sum_{k>=0} 2^k * (2*k+1)^n / k!.
a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^(k+1) * a(n-k).
Previous Showing 11-15 of 15 results.

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