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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).

%I A083007 #36 Jul 13 2023 04:56:12
%S A083007 0,1,-2,1,4,-5,-26,49,328,-809,-6710,20317,201772,-722813,-8370194,
%T A083007 34607305,457941136,-2145998417,-31945440878,167317266613,
%U A083007 2767413231220,-16020403322021,-291473080313162,1848020950359841,36679231132772824,-252778977216700025,-5435210060467425446
%N 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).
%H A083007 Seiichi Manyama, <a href="/A083007/b083007.txt">Table of n, a(n) for n = 0..510</a>
%H A083007 G. Almkvist and A. Meurman, <a href="https://mr.math.ca/article/values-of-bernoulli-polynomials-and-hurwitzs-zeta-function-at-rational-points/">Values of Bernoulli polynomials and Hurwitz's Zeta function at rational points</a>, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 104-109.
%H A083007 Ira M. Gessel, <a href="https://doi.org/10.5281/zenodo.7625111">On the Almkvist-Meurman Theorem for Bernoulli Polynomials</a>, Integers (2023) Vol. 23, #A14.
%H A083007 B. Sury, <a href="http://www.isibang.ac.in/~sury/bulllms.pdf">The value of Bernoulli Polynomials at rational numbers</a>, Bull. London Math. Soc. 25 (1993), 327-29.
%F A083007 E.g.f.: 3x/(1+e^x+e^(2x)). - _Ira M. Gessel_, Jan 28 2012
%F A083007 From _Peter Bala_, Mar 01 2015: (Start)
%F A083007 a(2*n+1) = (-1)^(n+1)*A002111(n) for n >= 1.
%F A083007 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.
%F A083007 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)
%p A083007 A083007 := proc(n)
%p A083007     3*x/(1+exp(x)+exp(2*x)) ;
%p A083007     coeftayl(%,x=0,n) ;
%p A083007     %*n! ;
%p A083007 end proc:
%p A083007 seq(A083007(n),n=0..30) ; # _R. J. Mathar_, Jul 13 2023
%t A083007 Range[0, 15]! CoefficientList[ Series[ 3x/(1 + Exp[x] + Exp[ 2x]), {x, 0, 15}], x] (* _Robert G. Wilson v_, Oct 26 2012 *)
%t A083007 Table[Sum[3^k BernoulliB[k]Binomial[n,k],{k,0,n-1}],{n,0,30}] (* _Harvey P. Dale_, May 26 2014 *)
%o A083007 (PARI) a(n)=sum(k=0,n-1,3^k*binomial(n,k)*bernfrac(k))
%Y A083007 Cf. A001469.
%Y A083007 Cf. A036968, A083008, A083009, A083010, A083011, A083012, A083013, A083014.
%Y A083007 Cf. A002111.
%K A083007 sign,easy
%O A083007 0,3
%A A083007 _Benoit Cloitre_, May 31 2003