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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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

A296787 Expansion of e.g.f. exp(x*arctan(x)) (even powers only).

Original entry on oeis.org

1, 2, 4, 24, -496, 36000, -3753408, 556961664, -111591202560, 29054584410624, -9541382573767680, 3858875286730168320, -1884995591107521540096, 1094305223336273239449600, -744771228363250138965196800, 587358379156469629707528929280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 20 2017

Keywords

Examples

			exp(x*arctan(x)) = 1 + 2*x^2/2! + 4*x^4/4! + 24*x^6/6! - 496*x^8/8! + ...

Crossrefs

Cf. A002019, A009252, A009273, A010050, A102059, A166356, A259647, A293192, A296788, A296789.

Programs

  • Mathematica
    nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTan[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 15; Table[(CoefficientList[Series[Exp[(I/2) x (Log[1 - I x] - Log[1 + I x])], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

Formula

a(n) = (2*n)! * [x^(2*n)] exp(x*arctan(x)).
a(n) ~ -(-1)^n * 2^(2*n-1) * n^(2*n-1) / exp(2*n). - Vaclav Kotesovec, Dec 21 2017
Showing 1-2 of 2 results.

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