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.
-
a:=series(sec(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019
-
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.
-
a:=series(sech(x*exp(x)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 27 2019
-
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]!
A331612
E.g.f.: exp(1 / (2 - sec(x)) - 1) (even powers only).
Original entry on oeis.org
1, 1, 14, 481, 30449, 3064306, 448104029, 89621046061, 23468873468054, 7786478152466221, 3190021872763911149, 1580829351026679822586, 931656913226081002622489, 643808850722810399312420281, 515431991397502094847830786174, 473171296200788822261644150349881
Offset: 0
-
nmax = 15; Table[(CoefficientList[Series[Exp[1/(2 - Sec[x]) - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
e[0] = 1; e[n_] := e[n] = (-1)^n (1 - Sum[(-1)^j Binomial[2 n, 2 j] 3^(2 (n - j)) e[j], {j, 0, n - 1}]); A002114[n_] := e[n]/2^(2 n + 1); a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] A002114[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
With[{nn=40},Take[CoefficientList[Series[Exp[1/(2-Sec[x])-1],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Aug 08 2023 *)
A302579
Expansion of e.g.f. exp(cosh(x)/cos(x)-1) (even powers only).
Original entry on oeis.org
1, 2, 24, 632, 28784, 1991552, 193410624, 24993180032, 4134783110144, 850499728758272, 212579274719007744, 63381008507902595072, 22200896917210834817024, 9019985888570141052280832, 4204783981520054371872374784, 2228007853953954434037178007552
Offset: 0
exp(cosh(x)/cos(x)-1) = 1 + 2*x^2/2! + 24*x^4/4! + 632*x^6/6! + 28784*x^8/8! + ...
-
nmax = 15; Table[(CoefficientList[Series[Exp[Cosh[x]/Cos[x] - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A331818
E.g.f.: exp(1 - sec(x)) (even powers only).
Original entry on oeis.org
1, -1, -2, -1, 253, 12854, 668053, 39148199, 2456262898, 130790155859, -3853032641387, -4312625669814166, -1531200886955161127, -489884373969089299201, -159097972223555719000922, -54064488830901650420384521, -19284261543086608770504566147
Offset: 0
-
nmax = 16; Table[(CoefficientList[Series[Exp[1 - Sec[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[2 n - 1, 2 k - 1] Abs[EulerE[2 k]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
-
a(n) = {polcoef(serlaplace(exp(1 - 1/cos(x + O(x^(2*n + 1))))), 2*n)} \\ Andrew Howroyd, Jan 27 2020
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