A176328 Numerators of the rational sequence with e.g.f. (x/2)*(exp(-x) + 1)/(exp(x) - 1).

1, -1, 7, -3, 59, -5, 127, -7, 119, -9, 335, -11, 15689, -13, 49, -15, 463, -17, 51049, -19, -171311, -21, 856031, -23, -236331331, -25, 8553181, -27, -23749448849, -29, 8615841490835, -31, -7709321033057, -33, 2577687858469

Offset: 0

Author

Paul Curtz, Apr 15 2010

Keywords

• frac
• sign

Comments

Numerator of the n-term of the inverse binomial transform of the modified Bernoulli sequence A176327(k)/A027642(k).

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.

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.

Examples

			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
		

Crossrefs

Cf. A176591 (denominators), A141056 (denominators for the unsigned variant).

Programs

Maple

read("transforms") ; evb := [1,0,seq(bernoulli(n),n=2..50)] ; BINOMIALi(evb) ; apply(numer,%) ; # R. J. Mathar, Dec 01 2010
seq(numer((-1)^n*(bernoulli(n,1)+bernoulli(n,2))/2),n=0..34); # Peter Luschny, Jun 17 2012
gf := cosh(x*z)*z/(1-exp(-z)): ser := series(gf, z, 35):
seq((-1)^n*numer(subs(x=1, n!*coeff(ser, z, n))), n=0..34); # Peter Luschny, Aug 19 2018

Mathematica

terms = 35; egf = (x/2)*((Exp[-x] + 1)/(Exp[x] - 1)) + O[x]^(terms);
CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* Jean-François Alcover, Jun 13 2017 *)

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

Formula

Conjecture: a(2*n+1) = -2*n-1.

a(n) = numerator((-1)^n*(bernoulli(n, 1) + bernoulli(n, 2))/2). - Peter Luschny, Jun 17 2012

(-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

Extensions

Apparently incorrect claims concerning the inverse binomial transform of the B_n removed by R. J. Mathar, Dec 01 2010

New name from Peter Luschny, Jun 17 2012