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A349264
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Generalized Euler numbers, a(n) = n!*[x^n](sec(4*x)*(sin(4*x) + 1)).
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23
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1, 4, 16, 128, 1280, 16384, 249856, 4456448, 90767360, 2080374784, 52975108096, 1483911200768, 45344872202240, 1501108249821184, 53515555843342336, 2044143848640217088, 83285910482761809920, 3605459138582973251584, 165262072909347030040576, 7995891855149741436305408
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OFFSET
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0,2
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LINKS
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EXAMPLE
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Exponential generating functions of generalized Euler numbers in context:
egf1 = sec(1*x)*(sin(x) + 1).
egf2 = sec(2*x)*(sin(x) + cos(x)).
egf3 = sec(3*x)*(sin(2*x) + cos(x)).
egf4 = sec(4*x)*(sin(4*x) + 1).
egf5 = sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x)).
egf6 = sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)).
egf7 = sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)).
egf8 = sec(8*x)*2*(sin(4*x) + cos(4*x)).
egf9 = sec(9*x)*(4*sin(3*x) + 2)*cos(3*x)^2.
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MAPLE
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sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);
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MATHEMATICA
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m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)
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PROG
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(PARI) seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(4*x) + 1)/cos(4*x)))} \\ Andrew Howroyd, Nov 20 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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