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A076552
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a(n) = (-1)^(n+1)/3/(2n+1)*sum(k=0,n,16^k*B(2k)*C(2n+1,2k)) where B(k) denotes the k-th Bernoulli number.
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2
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1, 1, 21, 461, 16841, 900921, 66453661, 6463837381, 801626558481, 123457062745841, 23116291464379301, 5171511387852362301, 1362357503097707964121, 417419880467876621822761, 147181297749674368184560941, 59173130526513096478888263221
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OFFSET
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1,3
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COMMENTS
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Terms are of form 10k+1.
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LINKS
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FORMULA
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Conjectural e.g.f. with offset 0 (checked up to a(14)): 1/3*(2 - cos(x)^2 + 2*cos(x)^4)/cos(x)^3 = 1 + x^2/2! + 21*x^4/4! + 461*x^6/6! + .... (End)
G.f.: 1/(Q(0)*3*x) + 2/(3*x^2*(1+x)) - 2/(3*x^2) + 1/(3*x), where Q(k) = 1 - x*(k+1)^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 19 2013
Conjecture: a(n) = -1/3*(-4)^n*E(2*n,-1/2), where E(n,x) is the n-th Euler polynomial. - Peter Bala, Sep 25 2016
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MATHEMATICA
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PROG
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(PARI) a(n)=(-1)^(n+1)/3/(2*n+1)*sum(k=0, n, 16^k*bernfrac(2*k)*binomial(2*n+1, 2*k))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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