This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A263968 #38 Feb 16 2025 08:33:27
%S A263968 -3,4,-18,112,-930,9664,-120498,1752832,-29140290,545004544,
%T A263968 -11325668178,258892951552,-6456024679650,174410345857024,
%U A263968 -5074158021135858,158168121299894272,-5258993667674555010,185786981314092335104,-6949466928081909755538
%N 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.
%C A263968 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).
%H A263968 G. C. Greubel, <a href="/A263968/b263968.txt">Table of n, a(n) for n = 0..100</a> [a(62) corrected by _Georg Fischer_, Jun 29 2021]
%H A263968 Daniele Parisse, <a href="https://math.colgate.edu/~integers/y70/y70.pdf">On hypersequences of an arbitrary sequence and their weighted sums</a>, Integers (2024) Vol. 24, Art. No. A70. See p. 26.
%H A263968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H A263968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>.
%F A263968 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).
%F A263968 a(n) = (-1)^(n+1)*(2*A000556(n) + A000557(n)).
%F A263968 E.g.f.: -(1+2*exp(x))/(1+2*sinh(x)).
%F A263968 a(n) ~ (-1)^(n+1) * n! / log((1+sqrt(5))/2)^(n+1). - _Vaclav Kotesovec_, Oct 31 2015
%e A263968 For n = 4, Li_{-4}(phi) = -930 - 416*sqrt(5), so a(4) = -930 and A000557(4) = 416.
%p A263968 a := n -> polylog(-n,(1+sqrt(5))/2)+polylog(-n,(1-sqrt(5))/2):
%p A263968 seq(round(evalf(a(n),32)), n=0..18); # _Peter Luschny_, Nov 01 2015
%t A263968 Round@Table[PolyLog[-n, GoldenRatio] + PolyLog[-n, 1-GoldenRatio], {n, 0, 20}]
%t A263968 Table[(-1)^(n+1) Sum[k! LucasL[k+2] StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
%o A263968 (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
%Y A263968 Cf. A000032, A000556, A000557, A048993.
%K A263968 sign
%O A263968 0,1
%A A263968 _Vladimir Reshetnikov_, Oct 30 2015