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A106233 An inverse Catalan transform of A003462.

%I A106233 #31 Sep 08 2022 08:45:17
%S A106233 0,1,3,5,5,0,-14,-41,-81,-121,-121,0,364,1093,2187,3281,3281,0,-9842,
%T A106233 -29525,-59049,-88573,-88573,0,265720,797161,1594323,2391485,2391485,
%U A106233 0,-7174454,-21523361,-43046721,-64570081,-64570081,0,193710244,581130733,1162261467
%N A106233 An inverse Catalan transform of A003462.
%C A106233 The g.f. is obtained from that of A003462 through the mapping g(x)->g(x(1-x)). A003462 may be retrieved through the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. Binomial transform of x(1+x)/(1+x^2+x^4).
%C A106233 The sequence is identical to its sixth differences. See A140344. - _Paul Curtz_, Nov 09 2012
%H A106233 Vincenzo Librandi, <a href="/A106233/b106233.txt">Table of n, a(n) for n = 0..1000</a>
%H A106233 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,6,-3).
%F A106233 G.f.: x(1-x)/((1-x+x^2)*(1-3*x+3*x^2));
%F A106233 a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(3^(n-k)-1)/2.
%F A106233 a(n) = Sum_{k=0..n} A109466(n,k)*A003462(k). - _Philippe Deléham_, Oct 30 2008
%F A106233 a(n) = (1/2)*[A057083(n) - [1,1,0,0,-1,-1]_6 ]. - _Ralf Stephan_, Nov 15 2010
%F A106233 a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4) = A140343(n+2) - A140343(n+1). - _Paul Curtz_, Nov 09 2012
%F A106233 a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - _Paul Curtz_, Nov 09 2012
%e A106233 From _Paul Curtz_, Nov 09 2012: (Start)
%e A106233 The sequence and its higher-order differences (periodic after 6 rows):
%e A106233    0,  1,  3,  5,  5,   0, -14, ...
%e A106233    1,  2,  2,  0, -5, -14, -27, ...
%e A106233    1,  0, -2, -5, -9, -13, -13, ...
%e A106233   -1, -2, -3, -4, -4,   0,  13, ...   = -A134581(n+1)
%e A106233   -1, -1, -1,  0,  4,  13,  27, ...
%e A106233    0,  0,  1,  4,  9,  14,  14, ...   = A140343(n+2)
%e A106233    0,  1,  3,  5,  5,   0, -14, ...
%e A106233 (End)
%t A106233 LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* _Vincenzo Librandi_, Dec 24 2018 *)
%o A106233 (Magma) I:=[0,1,3,5]; [n le 4 select I[n] else 4*Self(n-1)-7*Self(n-2)+ 6*Self(n-3)-3*Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 24 2018
%Y A106233 Cf. A103368.
%K A106233 easy,sign
%O A106233 0,3
%A A106233 _Paul Barry_, Apr 26 2005