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%I #36 Aug 19 2018 02:53:16
%S 1,-1,7,-3,59,-5,127,-7,119,-9,335,-11,15689,-13,49,-15,463,-17,51049,
%T -19,-171311,-21,856031,-23,-236331331,-25,8553181,-27,-23749448849,
%U -29,8615841490835,-31,-7709321033057,-33,2577687858469
%N Numerators of the rational sequence with e.g.f. (x/2)*(exp(-x) + 1)/(exp(x) - 1).
%C Numerator of the n-term of the inverse binomial transform of the modified Bernoulli sequence A176327(k)/A027642(k).
%C The sequence of modified Bernoulli numbers A176327(k)/A027642(k) is defined to be the same as the Bernoulli sequence, except the term at index k=1 which is zero.
%C Its inverse binomial transform is 1, -1, 7/6, -3/2, 59/30, -5/2, 127/42, -7/2, 119/30, -9/2, 335/66, -11/2, ...; the numerators define this sequence here.
%F Conjecture: a(2*n+1) = -2*n-1.
%F a(n) = numerator((-1)^n*(bernoulli(n, 1) + bernoulli(n, 2))/2. - _Peter Luschny_, Jun 17 2012
%F (-1)^n*a(n) are the numerators of the polynomials generated by cosh(x*z)*z/(1-exp(-z)) evaluated x=1 (see the example section). The denominators of these values are A141056. - _Peter Luschny_, Aug 18 2018
%e The first few of the polynomials mentioned in the formula section are: 1, 1/2, 1/6 + x^2, (3/2)*x^2, -1/30 + x^2 + x^4, (5/2)*x^4, 1/42 - (1/2)*x^2 +(5/2)*x^4 + x^6, (7/2)*x^6, -1/30 + (2/3)*x^2 - (7/3)*x^4 + (14/3)*x^6 + x^8, (9/2)*x^8, ... The values of these polynomials at x=1 start 1, 1/2, 7/6, 3/2, 59/30, 5/2, 127/42, 7/2, ... - _Peter Luschny_, Aug 18 2018
%p read("transforms") ; evb := [1,0,seq(bernoulli(n),n=2..50)] ; BINOMIALi(evb) ; apply(numer,%) ; # _R. J. Mathar_, Dec 01 2010
%p seq(numer((-1)^n*(bernoulli(n,1)+bernoulli(n,2))/2),n=0..34); # _Peter Luschny_, Jun 17 2012
%p gf := cosh(x*z)*z/(1-exp(-z)): ser := series(gf, z, 35):
%p seq((-1)^n*numer(subs(x=1, n!*coeff(ser, z, n))), n=0..34); # _Peter Luschny_, Aug 19 2018
%t terms = 35; egf = (x/2)*((Exp[-x] + 1)/(Exp[x] - 1)) + O[x]^(terms);
%t CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* _Jean-François Alcover_, Jun 13 2017 *)
%o (PARI) my(x = 'x + O('x^50)); apply(x->numerator(x), Vec(serlaplace((x/2)*(exp(-x) + 1)/(exp(x) - 1)))) \\ _Michel Marcus_, Aug 19 2018
%Y Cf. A176591 (denominators), A141056 (denominators for the unsigned variant).
%K frac,sign
%O 0,3
%A _Paul Curtz_, Apr 15 2010
%E Apparently incorrect claims concerning the inverse binomial transform of the B_n removed by _R. J. Mathar_, Dec 01 2010
%E New name from _Peter Luschny_, Jun 17 2012
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