A009121
Expansion of e.g.f. cosh(exp(x)*x).
Original entry on oeis.org
1, 0, 1, 6, 25, 100, 481, 2954, 20721, 151848, 1146721, 9111982, 77652169, 710421452, 6891125697, 69961213170, 738718169569, 8108554524112, 92647353941569, 1101958783026134, 13616813607795321, 174287243264606388, 2304515271134124577, 31424734896799742170
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cosh(x*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 26 2018
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a:= n-> add(`if`(k::odd, 0, binomial(n, k)*k^(n-k)), k=0..n):
seq(a(n), n=0..25); # Alois P. Heinz, Jan 15 2018
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With[{nn=30},CoefficientList[Series[Cosh[Exp[x]*x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 28 2015 *)
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first(n) = x='x+O('x^n); Vec(serlaplace(cosh(exp(x)*x))) \\ Iain Fox, Dec 23 2017
Definition clarified and prior Mathematica program replaced by
Harvey P. Dale, Dec 28 2015
A009448
E.g.f. sin(x*exp(x)).
Original entry on oeis.org
0, 1, 2, 2, -8, -84, -504, -2304, -6656, 15760, 484000, 5348320, 44393856, 288642368, 1137006976, -5372233216, -195241910272, -2961479980800, -34259767672320, -320473550808576, -2158264400250880, -1727938748171264, 278774033845987328
Offset: 0
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With[{nn=30},CoefficientList[Series[Sin[x Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 17 2012 *)
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a(n):=sum(binomial(n,2*j+1)*(2*j+1)^(n-2*j-1)*(-1)^j,j,0,(n-1)/2); /* Vladimir Kruchinin, Jun 10 2011 */
A009768
Expansion of e.g.f. tanh(exp(x)*x).
Original entry on oeis.org
0, 1, 2, 1, -20, -159, -594, 2465, 69560, 665665, 1593850, -67177791, -1413216540, -12990964063, 64480265318, 4811655319393, 90259507840240, 540272971703937, -20890652777843598, -798235260367432831, -12766815370452348580
Offset: 0
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With[{nn=20},CoefficientList[Series[Tanh[Exp[x]*x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 03 2023 *)
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my(x='x+O('x^30)); concat(0, Vec(serlaplace(tanh(exp(x)*x)))) \\ Michel Marcus, Oct 01 2021
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
A297010
Expansion of e.g.f. arcsinh(x*exp(x)).
Original entry on oeis.org
0, 1, 2, 2, -8, -76, -264, 1672, 36800, 261648, -1443680, -66164704, -792152448, 2482671424, 289529373056, 5294082629760, 1648955815936, -2474170098704128, -65494141255724544, -303927676523118080, 35926135133071923200, 1341060635191667045376
Offset: 0
arcsinh(x*exp(x)) = x^1/1! + 2*x^2/2! + 2*x^3/3! - 8*x^4/4! - 76*x^5/5! - 264*x^6/6! + ...
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a:=series(arcsinh(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[ArcSinh[x Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Log[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(asinh(exp(x)*x)), -n) \\ Iain Fox, Dec 23 2017
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 1-7 of 7 results.