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
A296439
Expansion of e.g.f. log(1 + arctanh(x))*exp(x).
Original entry on oeis.org
0, 1, 1, 4, 0, 53, -155, 2364, -15288, 216817, -2147215, 32932700, -433435816, 7431919285, -120703007451, 2326504612964, -44614898438480, 963118686971137, -21195404220321151, 508991484878443860, -12604990423335824688, 334199905021923072597, -9181752759370241656699, 266806716890671639953964
Offset: 0
E.g.f.: A(x) = x/1! + x^2/2! + 4*x^3/3! + 53*x^5/5! - 155*x^6/6! + 2364*x^7/7! - 15288*x^8/8! + ...
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a:=series(log(1+arctanh(x))*exp(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
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nmax = 23; CoefficientList[Series[Log[1 + ArcTanh[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Log[1 + (Log[1 + x] - Log[1 - 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 + atanh(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017
A294314
Expansion of e.g.f. log(1 + x*sec(x))*exp(x).
Original entry on oeis.org
0, 1, 1, 5, 0, 64, -245, 2757, -23576, 272256, -3270977, 45055845, -671589952, 10984688636, -193875825117, 3688182769117, -75085512079184, 1630385857436224, -37596306847103457, 917765946045581357, -23641953753495247624, 640958728426947233468, -18242640219843554954221
Offset: 0
log(1 + x*sec(x))*exp(x) = x/1! + x^2/2! + 5*x^3/3! + 64*x^5/5! - 245*x^6/6! + ...
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a:=series(log(1+x*sec(x))*exp(x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
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nmax = 22; CoefficientList[Series[Log[1 + x Sec[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
A296336
Expansion of e.g.f. log(1 + x*cos(x))*exp(x).
Original entry on oeis.org
0, 1, 1, -1, 0, -16, -5, 13, -1512, 12336, -98177, 444773, 1249160, -75244508, 1296229779, -15956614403, 130892534160, 59122803744, -31478991643617, 837166840532589, -14900078660739984, 181343532181629332, -480411914937891389, -54306065912649098539, 2080183542455901881320
Offset: 0
log(1 + x*cos(x))*exp(x) = x/1! + x^2/2! - x^3/3! - 16*x^5/5! - 5*x^6/6! + ...
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a:=series(log(1 + x*cos(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 + x Cos[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]!
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x='x+O('x^99); concat([0], Vec(serlaplace(exp(x)*log(1+x*cos(x))))) \\ Altug Alkan, Dec 28 2017
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