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%I #22 Mar 01 2020 04:25:15
%S 1,1,11,173,22931,1319183,233526463,29412432709,39959591850371,
%T 8797116290975003,4872532317019728133,1657631603843299234219,
%U 2718086236621937756966743,1321397724505770800453750299,1503342018433974345747514544039
%N Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n*euler(2*n)*x^n/(2*n)).
%C This sequence is related in a peculiar way to A223067, a sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes. See A280443 for more information.
%H Sergey Khrushchev, <a href="http://assets.cambridge.org/97805218/54191/frontmatter/9780521854191_frontmatter.pdf">Orthogonal Polynomials and Continued Fractions, From Euler's point of view</a>, Corollary 4.26, p. 192, 2008.
%F a(n) = numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n * euler(2*n)*x^n/(2*n)).
%F Let S = Sum_{n>=0} (-1)^n*euler(2*n)*x^n/(2*n) and w(n) = A005187(n) then a(n) = 2^w(n) * [x^n] exp(S). - _Peter Luschny_, Jan 05 2017
%p nmax:=14: f := series(exp(add((-1)^n*euler(2*n) * x^n/(2*n), n=1..nmax+1)), x=0, nmax+1): for n from 0 to nmax do a(n) := numer(coeff(f, x, n)) od: seq(a(n), n=0..nmax);
%o (Sage)
%o def A280442_list(prec):
%o P.<x> = PowerSeriesRing(QQ, default_prec=2*prec)
%o def g(x): return exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
%o R = P(g(x)).coefficients()
%o d = lambda n: 2^(2*n - sum(n.digits(2)))
%o return [d(n)*R[n] for n in (0..prec)]
%o print(A280442_list(14)) # _Peter Luschny_, Jan 05 2017
%Y Cf. A046161 (denominators).
%Y Cf. A000364 (Euler numbers), A223067, A255881, A280443.
%K nonn,frac,easy
%O 0,3
%A _Johannes W. Meijer_ and _Joseph Abate_, Jan 03 2017
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