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Expansion of 1 / ( (1+x)*(2*x+1)*(-1+2*x)^2 ).
1

%I #31 Sep 08 2022 08:45:34

%S 1,1,7,9,39,57,199,313,967,1593,4551,7737,20935,36409,94663,167481,

%T 422343,757305,1864135,3378745,8155591,14913081,35418567,65244729,

%U 152859079,283348537,656175559,1222872633,2803659207,5249404473

%N Expansion of 1 / ( (1+x)*(2*x+1)*(-1+2*x)^2 ).

%H Vincenzo Librandi, <a href="/A140787/b140787.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-4,-8).

%F a(n) = 2^n*(n/3 + 11/18) + (-1)^n* (2^(n-1) - 1/9).

%F a(2n) - 2a(2n-1) = A002450(n+1).

%F a(n) + a(n+1) = A134353(n+1). - _R. J. Mathar_, Nov 10 2013

%t max = 40; j[n_] := (2^n-(-1)^n)/3; jj = Table[{j[n], -j[n]}, {n, 0, max+2, 2}] // Flatten; a[0] = 1; a[n_] := a[n] = 2*a[n-1] + jj[[n+3]]; Table[a[n], {n, 0, max}] (* _Jean-François Alcover_, Sep 30 2013 *)

%o (Magma) [2^n*(n/3+11/18) + (-1)^n* (2^(n-1)-1/9): n in [0..40]]; // _Vincenzo Librandi_, Aug 08 2011

%Y Cf. A092808, A094359, A140505.

%K nonn,easy

%O 0,3

%A _Paul Curtz_, Jul 14 2008

%E Better name from _R. J. Mathar_, Jul 02 2011

%E Edited by _Ralf Stephan_, Nov 10 2013