A106233 An inverse Catalan transform of A003462.

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

Offset: 0

Author

Paul Barry, Apr 26 2005

Keywords

• easy
• sign

Comments

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

Examples

			From _Paul Curtz_, Nov 09 2012: (Start)
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)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

Crossrefs

Cf. A103368.

Programs

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

Mathematica

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 3, 5}, 35] (* Vincenzo Librandi, Dec 24 2018 *)

Formula

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) = Sum_{k=0..n} A109466(n,k)*A003462(k). - Philippe Deléham, Oct 30 2008

a(n) = (1/2)*[A057083(n) - [1,1,0,0,-1,-1]6 ]. - _Ralf Stephan, Nov 15 2010

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

a(n) is the binomial transform of the sequence 0, 1, 1, -1, -1, 0, ... = A103368(n+5). - Paul Curtz, Nov 09 2012