A321394 a(n) = (1/24)*n!*[x^n] (9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x)) where sectan(x) = sec(x) + tan(x).

1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, 87700736, 2220246387, 62010892800, 1892138207375, 62591994720256, 2230631475837767, 85188256574494720, 3470563987113896475, 150234341045137637376, 6886077311552162511547, 333165973379285030666240, 16967906593223743786978375

Offset: 0

Author

Peter Luschny, Nov 08 2018

Keywords

• nonn

Comments

See A320956 for motivation and definitions.

Crossrefs

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), this sequence (n=4), A320956.

Programs

Maple

sectan := x -> sec(x) + tan(x): # sin(Pi/4 + x/2)*csc(Pi/4 - x/2)
egf := 9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x):
ser := series(egf, x, 22): seq((1/24)*n!*coeff(ser, x, n), n=0..20);

Mathematica

m = 20;
sectan[x_] := Sec[x] + Tan[x];
egf = 9 + sectan[4x] + 6 sectan[2x] + 8 sectan[x];
(1/24) CoefficientList[egf + O[x]^(m+1), x] Range[0, m]! (* Jean-François Alcover, Aug 19 2021 *)

PARI

sectan(x) = 1/cos(x) + tan(x);
my(x='x+O('x^25)); Vec(serlaplace(9 + sectan(4*x) + 6*sectan(2*x) + 8*sectan(x)))/24 \\ Michel Marcus, Aug 19 2021