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 A000556 Expansion of exp(-x) / (1 - exp(x) + exp(-x)). (Formerly M3966 N1638) 2
 1, 1, 5, 31, 257, 2671, 33305, 484471, 8054177, 150635551, 3130337705, 71556251911, 1784401334897, 48205833997231, 1402462784186105, 43716593539939351, 1453550100421124417, 51350258701767067711, 1920785418183176050505 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS G. Ledin, Jr., On a certain kind of Fibonacci sums, Fib. Quart., 5 (1967), 45-58. Eric Weisstein's MathWorld, Polylogarithm. Eric Weisstein's MathWorld, Golden Ratio. Eric Weisstein's MathWorld, Lucas Number. FORMULA a(n) = Sum_{k=0..n}(k!*fibonacci(k+1)*stirling2(n, k). E.g.f.: 1/(1 + U(0)) where U(k)= 1 - 2^k/(1 - x/(x - (k+1)*2^k/U(k+1) ));(continued fraction 3rd kind, 3-step ). - Sergei N. Gladkovskii, Dec 05 2012 a(n) ~ 2*n! / ((5+sqrt(5)) * log((1+sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, May 04 2015 a(n) = (-1)^(n+1)*(Li_{-n}(1-phi)*phi + Li_{-n}(phi)/phi)/sqrt(5), where Li_n(x) is the polylogarithm, phi=(1+sqrt(5))/2 is the golden ratio. - Vladimir Reshetnikov, Oct 30 2015 MATHEMATICA CoefficientList[Series[E^(-x)/(1-E^x+E^(-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 04 2015 *) Round@Table[(-1)^(n+1) (PolyLog[-n, 1-GoldenRatio] GoldenRatio + PolyLog[-n, GoldenRatio]/GoldenRatio)/Sqrt[5], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 30 2015 *) PROG (PARI) a(n) = sum(k=0, n, k!*fibonacci(k+1)*stirling(n, k, 2)); \\ Michel Marcus, Oct 30 2015 CROSSREFS John W. Layman observes that this is also Sum (-2)^k*binomial(n, k)*b(n-k), where b() = A005923. Cf. A005923. Sequence in context: A126121 A167137 A279434 * A125598 A267436 A058892 Adjacent sequences:  A000553 A000554 A000555 * A000557 A000558 A000559 KEYWORD nonn,easy AUTHOR STATUS approved

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