This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349264 #30 Oct 30 2024 01:57:59
%S A349264 1,4,16,128,1280,16384,249856,4456448,90767360,2080374784,52975108096,
%T A349264 1483911200768,45344872202240,1501108249821184,53515555843342336,
%U A349264 2044143848640217088,83285910482761809920,3605459138582973251584,165262072909347030040576,7995891855149741436305408
%N A349264 Generalized Euler numbers, a(n) = n!*[x^n](sec(4*x)*(sin(4*x) + 1)).
%H A349264 Matthew House, <a href="/A349264/b349264.txt">Table of n, a(n) for n = 0..389</a>
%H A349264 William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, <a href="http://dx.doi.org/10.1090/S0025-5718-2011-02520-2">The generating function for the Dirichlet series Lm(s)</a>, Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.
%H A349264 Ruth Lawrence and Don Zagier, <a href="https://doi.org/10.4310/AJM.1999.v3.n1.a5">Modular forms and quantum invariants of 3-manifolds</a>, Asian J. Math. 3 (1999), no. 1, 93-107.
%H A349264 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967) 689-694.
%H A349264 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-68-99652-X">Corrigendum: Generalized Euler and class numbers</a>, Math. Comp. 22, (1968) 699.
%H A349264 D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%e A349264 Exponential generating functions of generalized Euler numbers in context:
%e A349264 egf1 = sec(1*x)*(sin(x) + 1).
%e A349264 [A000111, A000364, A000182]
%e A349264 egf2 = sec(2*x)*(sin(x) + cos(x)).
%e A349264 [A001586, A000281, A000464]
%e A349264 egf3 = sec(3*x)*(sin(2*x) + cos(x)).
%e A349264 [A007289, A000436, A000191]
%e A349264 egf4 = sec(4*x)*(sin(4*x) + 1).
%e A349264 [A349264, A000490, A000318]
%e A349264 egf5 = sec(5*x)*(sin(x) + sin(3*x) + cos(2*x) + cos(4*x)).
%e A349264 [A349265, A000187, A000320]
%e A349264 egf6 = sec(6*x)*(sin(x) + sin(5*x) + cos(x) + cos(5*x)).
%e A349264 [A001587, A000192, A000411]
%e A349264 egf7 = sec(7*x)*(-sin(2*x) + sin(4*x) + sin(6*x) + cos(x) + cos(3*x) - cos(5*x)).
%e A349264 [A349266, A064068, A064072]
%e A349264 egf8 = sec(8*x)*2*(sin(4*x) + cos(4*x)).
%e A349264 [A349267, A064069, A064073]
%e A349264 egf9 = sec(9*x)*(4*sin(3*x) + 2)*cos(3*x)^2.
%e A349264 [A349268, A064070, A064074]
%p A349264 sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);
%t A349264 m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* _Amiram Eldar_, Nov 20 2021 *)
%o A349264 (PARI) seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(4*x) + 1)/cos(4*x)))} \\ _Andrew Howroyd_, Nov 20 2021
%Y A349264 Row 4 of A349271.
%Y A349264 Cf. A000111, A001586, A007289, A349265, A001587, A349266, A349267, A349268.
%Y A349264 Secant bisections: A000364, A000281, A000436, A000490, A000187, A000192, A064068, A064069, A064070.
%Y A349264 Tangent bisections: A000182, A000464, A000191, A000318, A000320, A000411, A064072, A064073, A064074.
%Y A349264 Cf. A225147, A349429.
%K A349264 nonn
%O A349264 0,2
%A A349264 _Peter Luschny_, Nov 20 2021