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Generalized Euler numbers, a(n) = n!*[x^n](sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x))).
5

%I #16 Oct 30 2024 01:55:26

%S 2,4,30,272,3522,55744,1066590,23750912,604935042,17328937984,

%T 551609685150,19313964388352,737740947722562,30527905292468224,

%U 1360427147514751710,64955605537174126592,3308161927353377294082,179013508069217017790464,10256718523496425979562270

%N Generalized Euler numbers, a(n) = n!*[x^n](sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x))).

%C For references and examples see A349264.

%H Matthew House, <a href="/A349265/b349265.txt">Table of n, a(n) for n = 0..377</a>

%t m = 18; CoefficientList[Series[Sec[5*x] * (Sin[x] + Sin[3*x] + Cos[2*x] + Cos[4*x]), {x, 0, m}], x] * Range[0, m]! (* _Amiram Eldar_, Nov 20 2021 *)

%o (Sage)

%o t = PowerSeriesRing(QQ, 't', default_prec=19).gen()

%o f = (sin(t) + sin(3*t) + cos(2*t) + cos(4*t)) / cos(5*t)

%o f.egf_to_ogf().list()

%o (PARI) seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(x) + sin(3*x) + cos(2*x) + cos(4*x))/cos(5*x)))} \\ _Andrew Howroyd_, Nov 20 2021

%Y Row 5 of A349271.

%Y Cf. A000111, A001589, A007289, A349264, A001587.

%K nonn,changed

%O 0,1

%A _Peter Luschny_, Nov 20 2021