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 1-3 of 3 results.

A302609 a(n) = n! * [x^n] exp(n*x)*arctanh(x).

Original entry on oeis.org

0, 1, 4, 29, 288, 3649, 56160, 1017029, 21181440, 498682881, 13095232000, 379443829709, 12025239367680, 413761766695809, 15360425115176960, 611958601019294325, 26042588632355176448, 1179009749826940037889, 56579126414696034729984, 2868848293506101088635389
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Links

Crossrefs

Cf. A010050, A291484, A293193, A302583, A302584, A302585, A302586, A302587, A302605, A302606, A302608.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x] ArcTanh[x], {x, 0, n}], {n, 0, 19}]
nmax = 20; CoefficientList[Series[Log[(1 - LambertW[-x])/(1 + LambertW[-x])] / (2*(1 + LambertW[-x])), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 09 2019 *)

Formula

E.g.f.: log((1 - LambertW(-x))/(1 + LambertW(-x))) / (2*(1 + LambertW(-x))). - Vaclav Kotesovec, Jun 09 2019
a(n) ~ log(n) * n^n / 4 * (1 + (gamma + 3*log(2))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 09 2019
a(n) = Sum_{k=1..n} binomial(n,k)*(k-1)!*n^(n-k)*(1-(-1)^k)/2. - Fabian Pereyra, Oct 05 2024

A302606 a(n) = n! * [x^n] exp(n*x)*arcsinh(x).

Original entry on oeis.org

0, 1, 4, 26, 240, 2884, 42660, 748544, 15185856, 349574544, 9000902500, 256293989984, 7996078704240, 271246034903232, 9939835626507332, 391303051339622400, 16469438021801262848, 737992773619777599744, 35077254665501330210628, 1762671472887447792620032
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Links

Crossrefs

Cf. A001818, A002866, A291483, A302583, A302584, A302585, A302586, A302587, A302605, A302608, A302609.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x] ArcSinh[x], {x, 0, n}], {n, 0, 19}]

Formula

a(n) ~ arcsinh(1) * n^n = log(1 + sqrt(2)) * n^n. - Vaclav Kotesovec, Jun 09 2019
a(n) = Sum_{k=1..n, k odd} (-1)^((k-1)/2)*binomial(n,k)*(k-2)!!^2*n^(n-k). - Fabian Pereyra, Oct 05 2024

A302608 a(n) = n! * [x^n] exp(n*x)*arctan(x).

Original entry on oeis.org

0, 1, 4, 25, 224, 2649, 38880, 679449, 13749248, 315919665, 8122432000, 231002307449, 7199799644160, 244028744225993, 8936047251296256, 351569799174274425, 14789182545666244608, 662389019735008588129, 31470659616611382460416, 1580849762199983023572313
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Links

Crossrefs

Cf. A010050, A279927, A293192, A302583, A302584, A302585, A302586, A302587, A302605, A302606, A302609.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x] ArcTan[x], {x, 0, n}], {n, 0, 19}]
Join[{0}, Table[n^n (HypergeometricPFQ[{1, 1, 1 - n}, {2}, -(I/n)] + HypergeometricPFQ[{1, 1, 1 - n}, {2}, I/n])/2, {n, 19}]]

Formula

a(n) ~ arctan(1) * n^n. - Vaclav Kotesovec, Jun 09 2019
a(n) = Sum_{k=1..n, k odd} (-1)^((k-1)/2)*binomial(n,k)*(k-1)!*n^(n-k). - Fabian Pereyra, Oct 05 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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