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.

-3, 4, -18, 112, -930, 9664, -120498, 1752832, -29140290, 545004544, -11325668178, 258892951552, -6456024679650, 174410345857024, -5074158021135858, 158168121299894272, -5258993667674555010, 185786981314092335104, -6949466928081909755538

Offset: 0

Author

Vladimir Reshetnikov, Oct 30 2015

Keywords

• sign

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).

Examples

			For n = 4, Li_{-4}(phi) = -930 - 416*sqrt(5), so a(4) = -930 and A000557(4) = 416.
		

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.

Crossrefs

Cf. A000032, A000556, A000557, A048993.

Programs

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}]

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

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