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

0,3

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

Table of n, a(n) for n=0..14.

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

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

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);

Prog

(Sage)
def A280442_list(prec):
    P.<x> = 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

Crossrefs

Cf. A046161 (denominators).

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

Sequence in context: A230604 A161355 A223067 * A218330 A365034 A196664

Adjacent sequences: A280439 A280440 A280441 * A280443 A280444 A280445

Keyword

nonn,frac,easy

Author

Johannes W. Meijer and Joseph Abate, Jan 03 2017

Status

approved