A296728
Expansion of e.g.f. arcsin(x*cos(x)) (odd powers only).
Original entry on oeis.org
1, -2, -16, 8, 12672, 571264, -44351360, -12355211520, -452681248768, 478190483394560, 132554796040912896, -18854516962334277632, -27186884683859043123200, -5502410397289951851773952, 6273206188133923322747420672, 5389680791235134726930445369344
Offset: 0
arcsin(x*cos(x)) = x/1! - 2*x^3/3! - 16*x^5/5! + 8*x^7/7! + 12672*x^9/9! + ...
Cf.
A001818,
A009015,
A009016,
A009446,
A009447,
A009633,
A009634,
A012495,
A012780,
A101928,
A191512,
A296464,
A296466,
A296679,
A296680,
A296729,
A296730,
A296731,
A296740.
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nmax = 16; Table[(CoefficientList[Series[ArcSin[x Cos[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
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first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x*cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017
A296729
Expansion of e.g.f. arcsin(x*cosh(x)) (odd powers only).
Original entry on oeis.org
1, 4, 44, 1912, 156816, 21506816, 4420845376, 1271132964480, 487161448339712, 239980527068474368, 147742478026391141376, 111153314734461183924224, 100339775128577885016985600, 107037870347952811373977239552, 133204585741561810426003651444736
Offset: 0
arcsin(x*cosh(x)) = x/1! + 4*x^3/3! + 44*x^5/5! + 1912*x^7/7! + 156816*x^9/9! + ...
Cf.
A001818,
A009015,
A009016,
A009446,
A009447,
A009633,
A009634,
A012495,
A012780,
A101928,
A191512,
A296464,
A296466,
A296679,
A296680,
A296728,
A296730,
A296731,
A296740.
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nmax = 15; Table[(CoefficientList[Series[ArcSin[x Cosh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
-
first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(asin(x*cosh(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017
A101922
Numerators of expansion of e.g.f. 2^(-1/2) * arcsinh(cos(x)), even powers only.
Original entry on oeis.org
-1, -1, 11, 491, 11159, -460681, -103577629, -8160790429, 624333860399, 386787409545839, 68810049201689531, -6999828208693648549, -9872674440874152431161, -3255253386897615662908441, 346248578699462435167833491, 1072454627614122049417452882131
Offset: 1
-
Table[Numerator[(2n)!SeriesCoefficient[ArcSinh[Cos[x]]/Sqrt[2],{x,0,2n}]],{n,14}] (* Stefano Spezia, Aug 29 2022 *)
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lista(nn) = my(x='x + O('x^(nn+1)), p=serlaplace(asinh(cos(x))/sqrt(2))); vector(nn\2, k, round(polcoef(p, 2*k)*2^k)); \\ Michel Marcus, Sep 11 2022
A101924
Numerators of expansion of e.g.f. 2^(-1/2) * arccsch(cos(x)), even powers only.
Original entry on oeis.org
1, 7, 109, 3163, 166201, 14952367, 2002052389, 353291166403, 77829008955121, 21170264082173527, 7106489649576530269, 2913186117837522604843, 1426879448953133350342441, 816516326741659045770111487, 537701607855913139967684905749, 404270165862091267387117902574483
Offset: 1
-
Table[Numerator[(2n)!SeriesCoefficient[ArcCsch[Cos[x]]/Sqrt[2], {x,0,2n}]],{n,14}] (* Stefano Spezia, Aug 29 2022 *)
-
arccsch(x) = log((1+sqrt(x^2+1))/x);
lista(nn) = localprec(4*nn); my(x='x+O('x^(nn+1)), v=Vec((serlaplace(arccsch(cos(x))))/quadgen(8))); apply(round, vector(#v\2-1, k, v[2*k+1]*2^k)); \\ Michel Marcus, Sep 21 2022
Showing 1-4 of 4 results.
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