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A047859 a(n) = T(2, n), where T is the array given by A047858.

%I A047859 #39 Jan 03 2023 02:06:35
%S A047859 1,4,11,27,63,143,319,703,1535,3327,7167,15359,32767,69631,147455,
%T A047859 311295,655359,1376255,2883583,6029311,12582911,26214399,54525951,
%U A047859 113246207,234881023,486539263,1006632959,2080374783,4294967295,8858370047,18253611007,37580963839
%N A047859 a(n) = T(2, n), where T is the array given by A047858.
%C A047859 n-th difference of a(n), a(n-1), ..., a(0) is (3, 4, 5, ...).
%C A047859 From _Gus Wiseman_, Oct 14 2022: (Start)
%C A047859 Also the number of compositions of 2*(n+1) whose maximum part is n+1. These are compositions of 2*(n+1) whose maximum part equals the sum of their remaining parts. For example, the a(0) = 1 through a(2) = 11 compositions are:
%C A047859   (1,1)  (2,2)    (3,3)
%C A047859          (1,1,2)  (1,2,3)
%C A047859          (1,2,1)  (1,3,2)
%C A047859          (2,1,1)  (2,1,3)
%C A047859                   (2,3,1)
%C A047859                   (3,1,2)
%C A047859                   (3,2,1)
%C A047859                   (1,1,1,3)
%C A047859                   (1,1,3,1)
%C A047859                   (1,3,1,1)
%C A047859                   (3,1,1,1)
%C A047859 For length instead of maximum we have A001700.
%C A047859 These compositions are ranked by A357708. (End)
%H A047859 Vincenzo Librandi, <a href="/A047859/b047859.txt">Table of n, a(n) for n = 0..3000</a>
%H A047859 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,4).
%F A047859 Main diagonal of the array defined by: T(0, j) = j + 1 for j >= 0, T(i, 0) = i + 1 for i >= 0, T(i, j)= T(i-1, j-1) + T(i-1, j) + 1. a(n) = (n + 4)*2^(n-1) - 1. - _Benoit Cloitre_, Jun 17 2003
%F A047859 a(0) = 1, a(1) = 4, a(2) = 11, a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3). - _Vincenzo Librandi_, Sep 28 2011
%F A047859 G.f.: (1 - x - x^2)/((1 - x)*(1 - 2*x)^2). - _Colin Barker_, Aug 24 2016
%F A047859 a(n) = A045623(n) - 1. - _Gus Wiseman_, Oct 14 2022
%F A047859 E.g.f.: exp(x)*(exp(x)*(2 + x) - 1). - _Stefano Spezia_, Jan 02 2023
%o A047859 (Magma) [(n+4)*2^(n-1)-1: n in [0..30]]; // _Vincenzo Librandi_, Sep 28 2011
%o A047859 (PARI) Vec((1-x-x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ _Colin Barker_, Aug 24 2016
%Y A047859 A011782 counts compositions.
%Y A047859 Cf. A000120, A001511, A001700, A029931, A045623, A047858, A056239, A070939, A357708.
%K A047859 nonn,easy
%O A047859 0,2
%A A047859 _Clark Kimberling_