A285866 a(n) = numerator((-2)^n*Sum_{k=0..n} binomial(n,k) * Bernoulli(k, 1/2)).

1, -2, 11, -6, 127, -10, 221, -14, 367, -18, -1895, -22, 1447237, -26, -57253, -30, 118526399, -34, -5749677193, -38, 91546283957, -42, -1792042789427, -46, 1982765468376757, -50, -286994504449237, -54, 3187598676787485443, -58, -4625594554880206360895, -62

Offset: 0

Author

Wolfdieter Lang, May 03 2017

Keywords

• sign
• easy
• frac

Comments

Previous name: Numerators of alternating row sums of the rational triangle B2 = A285864/A285865.

The denominators are given in A141459.

Crossrefs

Cf. A027641/A027642, A141459, A285864/A285865.

See also A157781/A157782 and A157811/A285068.

Programs

Maple

a := n -> numer((-2)^n*add(binomial(n,k)*bernoulli(k,1/2), k=0..n)):
seq(a(n), n=0..31); # Peter Luschny, Jul 24 2020

Mathematica

a[n_] := (-2)^n Sum[Binomial[n, k] BernoulliB[k, 1/2], {k, 0, n}] // Numerator;
Table[a[n], {n, 0, 31}] (* Peter Luschny, Jul 24 2020 *)

SageMath

# uses [gen_bernoulli_number from A157811]
print([numerator((-1)^n*gen_bernoulli_number(n, 2)) for n in range(33)]) # Peter Luschny, Mar 26 2021

Formula

a(n) = numerator(Sum_{m=0..n} (-1)^m*A285864(n, m)/A285865(n, m)), n >= 0, where the rational triangle is B2(n, m) = binomial(m, m)*2^(n-m)*B(n-m), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).

Extensions

More terms from Indranil Ghosh, May 06 2017

New name by Peter Luschny, Jul 24 2020