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A132824 Row sums of triangle A132823.

%I A132824 #24 Apr 03 2025 14:30:12
%S A132824 1,2,2,4,10,24,54,116,242,496,1006,2028,4074,8168,16358,32740,65506,
%T A132824 131040,262110,524252,1048538,2097112,4194262,8388564,16777170,
%U A132824 33554384,67108814,134217676,268435402,536870856,1073741766,2147483588,4294967234,8589934528
%N A132824 Row sums of triangle A132823.
%H A132824 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A132824 Binomial transform of [1, 1, -1, 3, -1, 3, -1, 3, -1, 3, ...].
%F A132824 For n > 0, a(n) = 2 + 2^n - 2*n = 1 + A183155(n-1). - _R. J. Mathar_, Apr 04 2012
%F A132824 From _Colin Barker_, Jun 06 2014: (Start)
%F A132824 a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n > 3.
%F A132824 G.f.: -(4*x^3-x^2-2*x+1)/((x-1)^2*(2*x-1)). (End)
%F A132824 For n > 1, a(n) = A132732(n-1). - _Jeppe Stig Nielsen_, Dec 29 2017
%F A132824 From _Elmo R. Oliveira_, Apr 03 2025: (Start)
%F A132824 E.g.f.: exp(x)*(exp(x) - 2*(x - 1)) - 2.
%F A132824 a(n) = 2*A000325(n-1) for n >= 1. (End)
%e A132824 a(4) = 10 = sum of row 4 terms of triangle A132823: (1 + 2 + 4 + 2 + 1).
%e A132824 a(3) = 4 = (1, 3, 3, 1) dot (1, 1, -1, 3) = (1 + 3 -3 + 3).
%p A132824 A132824:=n->`if`(n=0, 1, 2+2^n-2*n); seq(A132824(n), n=0..30); # _Wesley Ivan Hurt_, Jun 06 2014
%t A132824 a[0] = 1; a[n_] := 2 + 2^n - 2*n; Table[a[n], {n, 0, 30}] (* _Wesley Ivan Hurt_, Jun 06 2014 *)
%Y A132824 Cf. A000325, A132823, A183155.
%Y A132824 Essentially the same as A132732.
%K A132824 nonn,easy
%O A132824 0,2
%A A132824 _Gary W. Adamson_, Sep 02 2007