OFFSET
0,2
LINKS
William Y. C. Chen, Neil J. Y. Fan, Jeffrey Y. T. Jia, The generating function for the Dirichlet series Lm(s), Mathematics of Computation, Vol. 81, No. 278, pp. 1005-1023, April 2012.
Ruth Lawrence and Don Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J. Math. 3 (1999), no. 1, 93-107.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967) 689-694.
D. Shanks, Corrigendum: Generalized Euler and class numbers, Math. Comp. 22, (1968) 699.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
EXAMPLE
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.
MAPLE
sec(4*x)*(sin(4*x) + 1): series(%, x, 20): seq(n!*coeff(%, x, n), n = 0..19);
MATHEMATICA
m = 19; CoefficientList[Series[Sec[4*x] * (Sin[4*x] + 1), {x, 0, m}], x] * Range[0, m]! (* Amiram Eldar, Nov 20 2021 *)
PROG
(PARI) seq(n)={my(x='x + O('x^(n+1))); Vec(serlaplace((sin(4*x) + 1)/cos(4*x)))} \\ Andrew Howroyd, Nov 20 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Nov 20 2021
STATUS
approved