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A110665 Sequence is {a(0,n)}, where a(m,0)=0, a(m,n) = a(m-1,n)+a(m,n-1) and a(0,n) is such that a(n,n) = n for all n.

%I A110665 #27 Sep 06 2024 08:05:50
%S A110665 0,1,0,-3,-4,0,6,7,0,-9,-10,0,12,13,0,-15,-16,0,18,19,0,-21,-22,0,24,
%T A110665 25,0,-27,-28,0,30,31,0,-33,-34,0,36,37,0,-39,-40,0,42,43,0,-45,-46,0,
%U A110665 48,49,0,-51,-52,0,54,55,0,-57,-58,0,60,61,0,-63,-64,0,66,67,0,-69,-70,0,72,73,0,-75,-76,0,78,79
%N A110665 Sequence is {a(0,n)}, where a(m,0)=0, a(m,n) = a(m-1,n)+a(m,n-1) and a(0,n) is such that a(n,n) = n for all n.
%H A110665 G. C. Greubel, <a href="/A110665/b110665.txt">Table of n, a(n) for n = 0..1000</a>
%H A110665 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,2,-1).
%F A110665 a(0, n) = n - Sum_{k=0..(n-1)} binomial(2*n-k-1, n-1)*a(0, k).
%F A110665 From _Franklin T. Adams-Watters_, May 12 2006: (Start)
%F A110665 a(n) = n * A010892(n), where A010892 is periodic sequence [1,1,0,-1,-1,0].
%F A110665 G.f.: (x-2*x^2)/(1-x+x^2)^2. (End)
%e A110665 a(0,n): 0, 1, 0, -3, -4,...
%e A110665 a(1,n): 0, 1, 1, -2, -6,...
%e A110665 a(2,n): 0, 1, 2, 0, -6,...
%e A110665 a(3,n): 0, 1, 3, 3, -3,...
%e A110665 a(4,n): 0, 1, 4, 7, 4,...
%e A110665 Main diagonal of array is 0, 1, 2, 3, 4,...
%t A110665 LinearRecurrence[{2,-3,2,-1},{0,1,0,-3},80] (* _Harvey P. Dale_, Dec 19 2015 *)
%t A110665 a[0, n_] := a[0, n] = If[n < 3, {0, 1, 0}[[n+1]], (n((n-2)a[0, n-1] - (n-1)a[0, n-2]))/((n-1)(n-2))];
%t A110665 Table[a[0, n], {n, 0, 79}] (* _Jean-François Alcover_, Mar 29 2020 *)
%o A110665 (PARI) my(x='x+O('x^50)); concat([0], Vec((x-2*x^2)/(1-x+x^2)^2)) \\ _G. C. Greubel_, Sep 03 2017
%Y A110665 Cf. A110666 - A110672.
%K A110665 easy,sign
%O A110665 0,4
%A A110665 _Leroy Quet_, Aug 02 2005
%E A110665 More terms from _Franklin T. Adams-Watters_, May 12 2006