A296438
Expansion of e.g.f. log(1 + arctan(x))*exp(x).
Original entry on oeis.org
0, 1, 1, 0, 0, 13, 5, -336, -56, 18593, -6735, -1598520, 1192664, 205475645, -255011835, -36324220856, 62049925040, 8519764352097, -18835422533375, -2551646722754512, 6927586371061712, 951619735931190157, -3077560879933239899, -432185107142832520576, 1624964470900980885432
Offset: 0
E.g.f.: A(x) = x/1! + x^2/2! + 13*x^5/5! + 5*x^6/6! - 336*x^7/7! - 56*x^8/8! + ...
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a:=series(log(1+arctan(x))*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
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nmax = 24; CoefficientList[Series[Log[1 + ArcTan[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Log[1 + I (Log[1 - I x] - Log[1 + I x])/2] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
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my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + atan(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017
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
Cf.
A010050,
A279927,
A293192,
A302583,
A302584,
A302585,
A302586,
A302587,
A302605,
A302606,
A302609.
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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}]]
A291484
Expansion of e.g.f. arctanh(x)*exp(x).
Original entry on oeis.org
0, 1, 2, 5, 12, 49, 190, 1301, 7224, 69441, 495898, 6095429, 53005700, 792143793, 8110146070, 142633278997, 1679413757168, 33964965659649, 451969255722162, 10331348137881349, 153288815339260796, 3907452790559751857, 63949589015139119598, 1798373345567005989781, 32179694275204166066728
Offset: 0
E.g.f.: A(x) = x/1! + 2*x^2/2! + 5*x^3/3! + 12*x^4/4! + 49*x^5/5! + ...
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a:=series(arctanh(x)*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
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nmax = 24; Range[0, nmax]! CoefficientList[Series[ArcTanh[x] Exp[x], {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Log[(1 + x)/(1 - x)] Exp[x]/2, {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Sum[x^(2 k + 1)/(2 k + 1), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n+1,2k+1](n-2k)/(n+1) (2 k)!, {k,0,n/2}],{n,0,12}] (* Emanuele Munarini, Dec 16 2017 *)
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makelist(sum(binomial(n+1,2*k+1)*(n-2*k)/(n+1)*(2*k)!,k,0,floor(n/2)),n,0,12); /* Emanuele Munarini, Dec 16 2017 */
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first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)*exp(x)), -n) \\ Iain Fox, Dec 16 2017
Showing 1-3 of 3 results.