A280442 Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n*euler(2*n)*x^n/(2*n)).

1, 1, 11, 173, 22931, 1319183, 233526463, 29412432709, 39959591850371, 8797116290975003, 4872532317019728133, 1657631603843299234219, 2718086236621937756966743, 1321397724505770800453750299, 1503342018433974345747514544039

Offset: 0

Author

Johannes W. Meijer and Joseph Abate, Jan 03 2017

Keywords

• nonn
• frac
• easy

Comments

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.

Links

• Sergey Khrushchev, Orthogonal Polynomials and Continued Fractions, From Euler's point of view, Corollary 4.26, p. 192, 2008.

Crossrefs

Cf. A046161 (denominators).

Cf. A000364 (Euler numbers), A223067, A255881, A280443.

Programs

Maple

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);

Sage

def A280442_list(prec):
    P. = PowerSeriesRing(QQ, default_prec=2*prec)
    def g(x): return exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    d = lambda n: 2^(2*n - sum(n.digits(2)))
    return [d(n)*R[n] for n in (0..prec)]
print(A280442_list(14)) # Peter Luschny, Jan 05 2017

Formula

a(n) = numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n * euler(2*n)*x^n/(2*n)).

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