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A213593 Stirling transform of the first kind of the Lucas numbers A000032.

%I A213593 #25 Sep 08 2022 08:46:02
%S A213593 2,1,2,-3,10,-45,250,-1645,12490,-107415,1031690,-10943955,127058690,
%T A213593 -1602400085,21812913650,-318763741725,4977247397650,-82695799908975,
%U A213593 1456703469048850,-27117356172328675,531930264143933050
%N A213593 Stirling transform of the first kind of the Lucas numbers A000032.
%H A213593 G. C. Greubel, <a href="/A213593/b213593.txt">Table of n, a(n) for n = 0..445</a>
%H A213593 Jiaqiang Pan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Pan/pan19.html">Matrix Decomposition of the Unified Generalized Stirling Numbers and Inversion of the Generalized Factorial Matrices</a>, Journal of Integer Sequences 15 (2012) #12.6.6
%F A213593 a(0)=2; for n=1,2,3, ..., a(n) = r*(r-1)*(r-2)*...*(r-n+1) + s*(s-1)*(s-2)*...*(s-n+1), where r=(1+sqrt(5))/2 and s=(1-sqrt(5))/2.
%F A213593 From _Vladimir Reshetnikov_, Oct 20 2015: (Start)
%F A213593 Let phi=(1+sqrt(5))/2.
%F A213593 a(n) = Gamma(2-phi)/Gamma(2-phi-n)+Gamma(1+phi)/Gamma(1+phi-n).
%F A213593 Recurrence: a(0)=2, a(1)=1, a(n+2) = (1+n-n^2)*a(n) - 2*n*a(n+1).
%F A213593 E.g.f.: (1+(x+1)^sqrt(5))/(x+1)^(1/phi).
%F A213593 (End)
%F A213593 a(n) ~ (-1)^n * n! * n^((sqrt(5)-3)/2) / Gamma(2/(1+sqrt(5))). - _Vaclav Kotesovec_, Oct 21 2015
%F A213593 a(n) = Sum_{k=1..n} Stirling1(n,k)*Lucas(k). - _G. C. Greubel_, Jul 06 2019
%e A213593 For n=4, a(4) = r*(r-1)*(r-2)*(r-3) + s*(s-1)*(s-2)*(s-3) = 10.
%p A213593 A000032 := proc(n)
%p A213593         combinat[fibonacci](n+1)+combinat[fibonacci](n-1) ;
%p A213593 end proc:
%p A213593 A213593 := proc(n)
%p A213593         add(combinat[stirling1](n,i)*A000032(i),i=0..n) ;
%p A213593 end proc:
%p A213593 seq(A213593(n),n=0..20) ; # _R. J. Mathar_, Jun 26 2012
%t A213593 Expand@FunctionExpand@Table[Gamma[2 - GoldenRatio]/Gamma[2 - GoldenRatio - n] + Gamma[1 + GoldenRatio]/Gamma[1 - n + GoldenRatio], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 20 2015 *)
%t A213593 Table[If[n==0, 2, Sum[StirlingS1[n, k]*LucasL[k], {k, n}]], {n, 0, 25}] (* _G. C. Greubel_, Jul 06 2019 *)
%o A213593 (PARI) lucas(n) = fibonacci(n+1) + fibonacci(n-1);
%o A213593 vector(25, n, n--; if(n==0, 2, sum(k=1,n, stirling(n,k,1)*lucas(k)) )) \\ _G. C. Greubel_, Jul 06 2019
%o A213593 (Magma) [2] cat [(&+[StirlingFirst(n,k)*Lucas(k): k in [1..n]]): n in [1..25]]; // _G. C. Greubel_, Jul 06 2019
%o A213593 (Sage) [2]+[sum((-1)^(n-k)*stirling_number1(n,k)*lucas_number2(k,1,-1) for k in (1..n)) for n in (1..25)] # _G. C. Greubel_, Jul 06 2019
%o A213593 (GAP) Concatenation([2], List([1..25], n-> Sum([1..n], k-> (-1)^(n-k)* Stirling1(n, k)*Lucas(1,-1,k)[2] ))) # _G. C. Greubel_, Jul 06 2019
%Y A213593 Cf. A000032 (Lucas numbers), the general-term formula of which is L(n) = r^n + s^n.
%Y A213593 Cf. A008275, A323620.
%K A213593 sign
%O A213593 0,1
%A A213593 _Jiaqiang Pan_, Jun 15 2012