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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
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]
Daniele Parisse, On hypersequences of an arbitrary sequence and their weighted sums, Integers (2024) Vol. 24, Art. No. A70. See p. 26.
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
KEYWORD
sign
AUTHOR
STATUS
approved