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A083007
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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).
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12
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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)
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MAPLE
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3*x/(1+exp(x)+exp(2*x)) ;
coeftayl(%, x=0, n) ;
%*n! ;
end proc:
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=sum(k=0, n-1, 3^k*binomial(n, k)*bernfrac(k))
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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