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 A263968 a(n) = Li_{-n}(phi) + Li_{-n}(1-phi), where Li_n(x) is the polylogarithm, phi=(1+sqrt(5))/2 is the golden ratio. 2
 -3, 4, -18, 112, -930, 9664, -120498, 1752832, -29140290, 545004544, -11325668178, 258892951552, -6456024679650, 174410345857024, -5074158021135858, 158168121299894272, -5258993667674555010, 185786981314092335104, -6949466928081909755538 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 2*Li_{-n}(phi) = a(n) - (-1)^n*A000557(n)*sqrt(5), so a(n) represents integer terms in 2*Li_{-n}(phi), and A000557(n) (with alternating signs) represents terms proportional to sqrt(5). LINKS G. C. Greubel, Table of n, a(n) for n = 0..100 [a(62) corrected by Georg Fischer, Jun 29 2021] Eric Weisstein's World of Mathematics, Polylogarithm. Eric Weisstein's World of Mathematics, Golden Ratio. FORMULA a(n) = (-1)^(n+1)*Sum_{k=0..n} k!*Lucas(k+2)*Stirling2(n,k), where Lucas(n) = A000032(n) and A048993(n,k) = Stirling2(n,k). a(n) = (-1)^(n+1)*(2*A000556(n) + A000557(n)). E.g.f.: -(1+2*exp(x))/(1+2*sinh(x)). a(n) ~ (-1)^(n+1) * n! / log((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Oct 31 2015 EXAMPLE For n = 4, Li_{-4}(phi) = -930 - 416*sqrt(5), so a(4) = -930 and A000557(4) = 416. MAPLE a := n -> polylog(-n, (1+sqrt(5))/2)+polylog(-n, (1-sqrt(5))/2): seq(round(evalf(a(n), 32)), n=0..18); # Peter Luschny, Nov 01 2015 MATHEMATICA Round@Table[PolyLog[-n, GoldenRatio] + PolyLog[-n, 1-GoldenRatio], {n, 0, 20}] Table[(-1)^(n+1) Sum[k! LucasL[k+2] StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] PROG (PARI) vector(100, n, n--; (-1)^(n+1)*sum(k=0, n, k!*stirling(n, k, 2)*(2*fibonacci(k+1) + fibonacci(k+2)))) \\ Altug Alkan, Oct 31 2015 CROSSREFS Cf. A000032, A000556, A000557, A048993. Sequence in context: A254201 A222795 A318419 * A020344 A344459 A348349 Adjacent sequences:  A263965 A263966 A263967 * A263969 A263970 A263971 KEYWORD sign AUTHOR Vladimir Reshetnikov, Oct 30 2015 STATUS approved

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Last modified July 6 07:04 EDT 2022. Contains 355108 sequences. (Running on oeis4.)