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A026384 a(n) = Sum_{j=0..i, i=0..n} T(i,j), where T is the array in A026374.

%I A026384 #27 May 31 2023 13:08:02
%S A026384 1,3,8,18,43,93,218,468,1093,2343,5468,11718,27343,58593,136718,
%T A026384 292968,683593,1464843,3417968,7324218,17089843,36621093,85449218,
%U A026384 183105468,427246093,915527343,2136230468,4577636718,10681152343,22888183593,53405761718
%N A026384 a(n) = Sum_{j=0..i, i=0..n} T(i,j),  where T is the array in A026374.
%C A026384 Partial sums of A026383. Number of lattice paths from (0,0) that do not go to right of the line x=n, using the steps U=(1,1), D=(1,-1) and, at levels ...,-4,-2,0,2,4,..., also H=(2,0). Example: a(2)=8 because we have the empty path, U, D, UU, UD, DD, DU and H. - _Emeric Deutsch_, Feb 18 2004
%H A026384 Colin Barker, <a href="/A026384/b026384.txt">Table of n, a(n) for n = 0..1000</a>
%H A026384 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5).
%F A026384 G.f.: (1+2*x) / ((1-x)*(1-5*x^2)). - _Ralf Stephan_, Apr 30 2004
%F A026384 From _Colin Barker_, Nov 25 2016: (Start)
%F A026384 a(n) = (7*5^(n/2) - 3)/4 for n even.
%F A026384 a(n) = 3*(5^((n+1)/2) - 1)/4 for n odd.
%F A026384 a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n>2.
%F A026384 (End)
%p A026384 a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=5*a[n-2]+3 od: seq(a[n], n=1..29); # _Zerinvary Lajos_, Mar 17 2008
%t A026384 CoefficientList[Series[(1 + 2 x) / ((1 - x) (1 - 5 x^2)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 09 2017 *)
%t A026384 LinearRecurrence[{1,5,-5},{1,3,8},40] (* _Harvey P. Dale_, May 31 2023 *)
%o A026384 (PARI) Vec((2*x + 1)/(5*x^3 - 5*x^2 - x + 1) + O(x^40)) \\ _Colin Barker_, Nov 25 2016
%o A026384 (Magma) I:=[1,3,8]; [n le 3 select I[n] else Self(n-1)+5*Self(n-2)-5*Self(n-3): n in [1..35]]; // _Vincenzo Librandi_, Aug 09 2017
%Y A026384 Cf. A026383.
%K A026384 nonn,easy
%O A026384 0,2
%A A026384 _Clark Kimberling_