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A106233
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An inverse Catalan transform of A003462.
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4
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0, 1, 3, 5, 5, 0, -14, -41, -81, -121, -121, 0, 364, 1093, 2187, 3281, 3281, 0, -9842, -29525, -59049, -88573, -88573, 0, 265720, 797161, 1594323, 2391485, 2391485, 0, -7174454, -21523361, -43046721, -64570081, -64570081, 0, 193710244, 581130733, 1162261467
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OFFSET
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0,3
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COMMENTS
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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).
The sequence is identical to its sixth differences. See A140344. - Paul Curtz, Nov 09 2012
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LINKS
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FORMULA
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G.f.: x(1-x)/((1-x+x^2)*(1-3*x+3*x^2));
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(3^(n-k)-1)/2.
a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - Paul Curtz, Nov 09 2012
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EXAMPLE
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The sequence and its higher-order differences (periodic after 6 rows):
0, 1, 3, 5, 5, 0, -14, ...
1, 2, 2, 0, -5, -14, -27, ...
1, 0, -2, -5, -9, -13, -13, ...
-1, -2, -3, -4, -4, 0, 13, ... = -A134581(n+1)
-1, -1, -1, 0, 4, 13, 27, ...
0, 0, 1, 4, 9, 14, 14, ... = A140343(n+2)
0, 1, 3, 5, 5, 0, -14, ...
(End)
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MATHEMATICA
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LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* Vincenzo Librandi, Dec 24 2018 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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