A296979
Expansion of e.g.f. arcsin(log(1 + x)).
Original entry on oeis.org
0, 1, -1, 3, -12, 68, -480, 4144, -42112, 494360, -6581880, 98079696, -1617373296, 29245459176, -575367843960, 12235339942344, -279650131845120, 6836254328079936, -177979145883651648, 4916243253642325056, -143602294106947553280, 4422411460743707222784
Offset: 0
arcsin(log(1 + x)) = x^1/1! - x^2/2! + 3*x^3/3! - 12*x^4/4! + 68*x^5/5! - 480*x^6/6! + ...
Cf.
A001710,
A001818,
A003703,
A003708,
A009024,
A009454,
A009775,
A104150,
A189815,
A296980,
A296981,
A296982.
-
a:=series(arcsin(log(1+x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
-
nmax = 21; CoefficientList[Series[ArcSin[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I Log[1 + x] + Sqrt[1 - Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!
A296980
Expansion of e.g.f. arcsinh(log(1 + x)).
Original entry on oeis.org
0, 1, -1, 1, 0, -2, -30, 446, -3248, 12412, 16020, -211356, -10756944, 284038272, -3556910448, 19122463296, 135073768320, -1286054192304, -108801241372368, 3952903127312016, -65667347037774720, 339816855220730784, 8862271481944986336
Offset: 0
arcsinh(log(1 + x)) = x^1/1! - x^2/2! + x^3/3! - 2*x^5/5! - 30*x^6/6! + ...
Cf.
A001710,
A001818,
A003703,
A003708,
A009024,
A009454,
A009775,
A104150,
A296435,
A296979,
A296981,
A296982.
-
a:=series(arcsinh(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
-
nmax = 22; CoefficientList[Series[ArcSinh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Log[Log[1 + x] + Sqrt[1 + Log[1 + x]^2]], {x, 0, nmax}], x] Range[0, nmax]!
A296981
Expansion of e.g.f. arctan(log(1 + x)).
Original entry on oeis.org
0, 1, -1, 0, 6, -22, -30, 952, -5656, -9952, 508320, -3874992, -20690208, 833780400, -7697940432, -52230156288, 2467649024640, -24686997151104, -329724479772288, 14493628861307136, -159114034671287040, -2682505451050592256, 126421889770129637376
Offset: 0
arctan(log(1 + x)) = x^1/1! - x^2/2! + 6*x^4/4! - 22*x^5/5! - 30*x^6/6! + ...
Cf.
A001710,
A003703,
A003708,
A009024,
A009454,
A009775,
A010050,
A104150,
A110708,
A296979,
A296980,
A296982.
-
a:=series(arctan(log(1+x)),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
-
nmax = 22; CoefficientList[Series[ArcTan[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[(I/2) Log[1 - I Log[1 + x]] - (I/2) Log[1 + I Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
A296982
Expansion of e.g.f. arctanh(log(1 + x)).
Original entry on oeis.org
0, 1, -1, 4, -18, 118, -930, 8888, -98504, 1248784, -17790480, 281590032, -4901447232, 93064850448, -1914144990576, 42396742460928, -1006101059149440, 25466710774651776, -684902462140798848, 19503187752732408576, -586221766070655432960
Offset: 0
arctanh(log(1 + x)) = x^1/1! - x^2/2! + 4*x^3/3! - 18*x^4/4! + 118*x^5/5! - 930*x^6/6! + ...
Cf.
A001710,
A003703,
A003708,
A009024,
A009454,
A009775,
A010050,
A104150,
A202139,
A296979,
A296980,
A296981.
-
a:=series(arctanh(log(1+x)),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019
-
nmax = 20; CoefficientList[Series[ArcTanh[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Log[1 + Log[1 + x]]/2 - Log[1 - Log[1 + x]]/2, {x, 0, nmax}], x] Range[0, nmax]!
A306336
Expansion of e.g.f. sec(log(1 + x)) + tan(log(1 + x)).
Original entry on oeis.org
1, 1, 0, 1, -2, 10, -50, 320, -2340, 19640, -184900, 1932500, -22187200, 277576000, -3757884000, 54732418000, -853278998000, 14176686784000, -250046057846000, 4665989766386000, -91838330641200000, 1901405069222360000, -41307212202493120000, 939523370329035440000, -22327292561388519640000
Offset: 0
-
a:=series(sec(log(1 + x)) + tan(log(1 + x)),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 26 2019
-
nmax = 24; CoefficientList[Series[Sec[Log[1 + x]] + Tan[Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
e[n_] := e[n] = (2 I)^n If[EvenQ[n], EulerE[n, 1/2], EulerE[n, 0] I]; a[n_] := a[n] = Sum[StirlingS1[n, k] e[k], {k, 0, n}]; Table[a[n], {n, 0, 24}]
-
from itertools import accumulate
from sympy.functions.combinatorial.numbers import stirling
def A306336(n): # generator of terms
if n == 0: return 1
blist, c = (0,1), 0
for k in range(1,n+1):
c += stirling(n,k,kind=1,signed=True)*blist[-1]
blist = tuple(accumulate(reversed(blist),initial=0))
return c # Chai Wah Wu, Apr 18 2023
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