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]!
A331978
E.g.f.: -log(2 - cosh(x)) (even powers only).
Original entry on oeis.org
0, 1, 4, 46, 1114, 46246, 2933074, 263817646, 31943268634, 5009616448246, 987840438629794, 239217148602642046, 69790939492563608554, 24143849395162438623046, 9772368696995766705116914, 4575221153658910691872135246, 2453303387149157947685779986874
Offset: 0
-
ptan := proc(n) option remember;
if irem(n, 2) = 0 then 0 else
add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1, 2) fi end:
A331978 := n -> ptan(2*n - 1):
seq(A331978(n), n = 0..16); # Peter Luschny, Jun 06 2022
-
nmax = 16; Table[(CoefficientList[Series[-Log[2 - Cosh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A357693
Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).
Original entry on oeis.org
1, 0, -2, 6, -18, 60, -216, 756, -1620, -14256, 349272, -5452920, 78885576, -1143659088, 17074183104, -265437239760, 4316991698448, -73572489226368, 1314108286270560, -24584195654596512, 481215937895868384, -9843358555320333120, 210128893733994567552
Offset: 0
-
With[{nn=30},CoefficientList[Series[Cos[Sqrt[2]Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2024 *)
-
my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*log(1+x)))))
-
a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 1));
-
a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(2)*I+k)+prod(k=0, n-1, -sqrt(2)*I+k)))/2;
-
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=2, n, v[i+1]=-(2*i-3)*v[i]-(i^2-4*i+6)*v[i-1]); v;