A297009
Expansion of e.g.f. arcsin(x*exp(x)).
Original entry on oeis.org
0, 1, 2, 4, 16, 104, 816, 7792, 89216, 1177920, 17603200, 294334976, 5442281472, 110221745152, 2426850793472, 57718658411520, 1474590580228096, 40274407232294912, 1171043235561185280, 36115912820342407168, 1177554628069200035840, 40471207964013864124416
Offset: 0
arcsin(x*exp(x)) = x^1/1! + 2*x^2/2! + 4*x^3/3! + 16*x^4/4! + 104*x^5/5! + 816*x^6/6! + ...
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a:=series(arcsin(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019
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nmax = 21; CoefficientList[Series[ArcSin[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-I Log[I x Exp[x] + Sqrt[1 - x^2 Exp[2 x]]], {x, 0, nmax}], x] Range[0, nmax]!
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first(n) = my(x='x+O('x^n)); Vec(serlaplace(asin(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017
A380050
E.g.f. A(x) satisfies A(x) = sqrt( 1 + 2*x*exp(x)*A(x) ).
Original entry on oeis.org
1, 1, 3, 9, 25, 25, -429, -4151, -8175, 320625, 5241475, 23329801, -705579159, -18521117303, -150119840493, 3366485315145, 138253031778721, 1780881865542625, -28047359274759549, -1854674541474191351, -34985197604145203655, 332608115115937927161
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(asinh(x*exp(x)))))
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a(n) = n!*sum(k=0, n, 2^k*k^(n-k)*binomial(k/2+1/2, k)/((k+1)*(n-k)!));
A294312
Expansion of e.g.f. sec(x*exp(x)).
Original entry on oeis.org
1, 0, 1, 6, 29, 180, 1501, 14434, 154265, 1856232, 24953401, 368767102, 5936244533, 103519338780, 1944554725205, 39134556793050, 840024295910833, 19157944025344464, 462629389438242673, 11792248121970820598, 316398168231432879565, 8913743651504295251844
Offset: 0
sec(x*exp(x)) = 1 + x^2/2! + 6*x^3/3! + 29*x^4/4! + 180*x^5/5! + 1501*x^6/6! + ...
Cf.
A000364,
A009007,
A009017,
A009121,
A009300,
A009448,
A009565,
A009635,
A009768,
A139134,
A191719,
A216401,
A217502,
A294313,
A297009,
A297010.
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a:=series(sec(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019
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nmax = 21; CoefficientList[Series[Sec[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/Cos[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
A294313
Expansion of e.g.f. sech(x*exp(x)).
Original entry on oeis.org
1, 0, -1, -6, -19, 20, 899, 7966, 27705, -366552, -8374201, -80690302, 9794597, 16015845820, 317370642315, 2554368906150, -37571987331343, -1784464543440304, -31315944840101233, -80221319702865398, 12685422355781995485, 422083364962616527716
Offset: 0
sech(x*exp(x)) = 1 - x^2/2! - 6*x^3/3! - 19*x^4/4! + 20*x^5/5! + 899*x^6/6! + ...
Cf.
A000364,
A009017,
A009121,
A009301,
A009448,
A009565,
A009635,
A009768,
A191719,
A216401,
A217502,
A294312,
A296544,
A297009,
A297010.
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a:=series(sech(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019
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nmax = 21; CoefficientList[Series[Sech[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/Cosh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-4 of 4 results.