A083007 a(n) = Sum_{k=0..n-1} 3^k*B(k)*C(n,k) where B(k) is the k-th Bernoulli number and C(n,k)=binomial(n,k).

0, 1, -2, 1, 4, -5, -26, 49, 328, -809, -6710, 20317, 201772, -722813, -8370194, 34607305, 457941136, -2145998417, -31945440878, 167317266613, 2767413231220, -16020403322021, -291473080313162, 1848020950359841, 36679231132772824, -252778977216700025, -5435210060467425446

Offset: 0

Author

Benoit Cloitre, May 31 2003

Keywords

• sign
• easy

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.

Crossrefs

Cf. A001469.

Cf. A036968, A083008, A083009, A083010, A083011, A083012, A083013, A083014.

Cf. A002111.

Programs

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 *)

PARI

a(n)=sum(k=0,n-1,3^k*binomial(n,k)*bernfrac(k))

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)