OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..510
G. Almkvist and A. Meurman, Values of Bernoulli polynomials and Hurwitz's Zeta function at rational points, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 104-109.
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.
B. Sury, The value of Bernoulli Polynomials at rational numbers, Bull. London Math. Soc. 25 (1993), 327-29.
FORMULA
E.g.f.: 3x/(1+e^x+e^(2x)). - Ira M. Gessel, Jan 28 2012
From Peter Bala, Mar 01 2015: (Start)
a(2*n+1) = (-1)^(n+1)*A002111(n) for n >= 1.
a(n) = 3^n * ( B(n,1/3) - B(n,0) ), where B(n,x) denotes the n-th Bernoulli polynomial. More generally, Almkvist and Meurman show that k^n * ( B(n, 1/k) - B(n, 0) ) is an integer sequence for k = 2,3,4,..., which proves the integrality of A083008 through A083014.
a(0) = 1 and for n >= 1, a(n) = 1 - 1/(n + 1)*Sum_{k = 1..n-1} 3^(n-k)*binomial(n+1,k)*a(k) (Sury, Section 1). (End)
MAPLE
A083007 := proc(n)
3*x/(1+exp(x)+exp(2*x)) ;
coeftayl(%, x=0, n) ;
%*n! ;
end proc:
seq(A083007(n), n=0..30) ; # R. J. Mathar, Jul 13 2023
MATHEMATICA
Range[0, 15]! CoefficientList[ Series[ 3x/(1 + Exp[x] + Exp[ 2x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Table[Sum[3^k BernoulliB[k]Binomial[n, k], {k, 0, n-1}], {n, 0, 30}] (* Harvey P. Dale, May 26 2014 *)
PROG
(PARI) a(n)=sum(k=0, n-1, 3^k*binomial(n, k)*bernfrac(k))
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Benoit Cloitre, May 31 2003
STATUS
approved