A302542
Expansion of e.g.f. arctan(x)/cos(x) (odd powers only).
Original entry on oeis.org
1, 1, 29, -139, 31737, -1824151, 313750293, -51584719523, 13137192234225, -3947317975733039, 1522475446731094285, -702509124781480897211, 389722900767594460770025, -253710144786166583863030983, 192285396891961478711402819077, -167564604997707653568802119363795
Offset: 0
arctan(x)/cos(x) = x/1! + x^3/3! + 29*x^5/5! - 139*x^7/7! + 31737*x^9/9! - 1824151*x^11/11! + ...
Cf.
A000182,
A000364,
A000795,
A002084,
A003701,
A003702,
A010050,
A012801,
A191003,
A301942,
A302444,
A302543.
-
nmax = 16; Table[(CoefficientList[Series[ArcTan[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
A302543
Expansion of e.g.f. arctanh(x)/cos(x) (odd powers only).
Original entry on oeis.org
1, 5, 69, 2001, 104073, 8723549, 1088372557, 190057979177, 44285819490065, 13267464006201781, 4964113699657822805, 2266816666007859759489, 1239999748307938170531225, 800189083150907165762837517, 601369618369661775955962338653, 520607107122686183781743903500505
Offset: 0
arctanh(x)/cos(x) = x/1! + 5*x^3/3! + 69*x^5/5! + 2001*x^7/7! + 104073*x^9/9! + ...
Cf.
A000182,
A000364,
A000795,
A002084,
A003701,
A003702,
A010050,
A012840,
A296743,
A301942,
A302444,
A302542.
-
nmax = 16; Table[(CoefficientList[Series[ArcTanh[x]/Cos[x], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
A111196
a(n) = 2^(-n)*Sum_{k=0..n} binomial(2*n+1, 2*k+1)*A000364(n-k).
Original entry on oeis.org
1, 2, 9, 78, 1141, 25442, 804309, 34227438, 1886573641, 130746521282, 11127809595009, 1141012634368398, 138730500808639741, 19735099323279743522, 3247323803322747092109, 611982206046097666022958
Offset: 0
-
t = Range[0, 32]!CoefficientList[ Series[ Sec[x], {x, 0, 32}], x]; f[n_] := 2^(-n)*Sum [Binomial[2n + 1, 2k + 1]*t[[2n - 2k + 1]], {k, 0, n}]; Table[ f[n], {n, 0, 16}] (* Robert G. Wilson v, Oct 24 2005 *)
Table[Sum[Binomial[2*n + 1, 2*k + 1]*Abs[EulerE[2*(n-k)]], {k, 0, n}] / 2^n, {n, 0, 20}] (* Vaclav Kotesovec, Jul 10 2021 *)
A336024
Expansion of e.g.f. (1 + sinh(x)) / cos(x).
Original entry on oeis.org
1, 1, 1, 4, 5, 36, 61, 624, 1385, 18256, 50521, 814144, 2702765, 51475776, 199360981, 4381112064, 19391512145, 482962852096, 2404879675441, 66942218896384, 370371188237525, 11394877025289216, 69348874393137901, 2336793875186479104, 15514534163557086905
Offset: 0
-
nmax = 24; CoefficientList[Series[(1 + Sinh[x])/Cos[x], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := If[EvenQ[n], Abs[EulerE[n]], Sum[Binomial[n, k] Abs[EulerE[k]], {k, 0, n}]]; Table[a[n], {n, 0, 24}]
A352906
Expansion of e.g.f. sinh(x) / (1 - sin(x)).
Original entry on oeis.org
0, 1, 2, 7, 24, 101, 472, 2507, 14784, 96361, 687392, 5332207, 44694144, 402663821, 3880880512, 39848805107, 434306095104, 5007757446481, 60907946680832, 779345606053207, 10465549612529664, 147168296199468341, 2162785172079204352, 33155700678534788507, 529311396083558989824
Offset: 0
-
nmax = 24; CoefficientList[Series[Sinh[x]/(1 - Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!