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A244213 Inverse binomial transform of -2 followed by A000032(n+1). 1

%I #19 Jan 14 2023 16:52:38

%S -2,3,-1,0,3,-7,14,-25,43,-72,119,-195,318,-517,839,-1360,2203,-3567,

%T 5774,-9345,15123,-24472,39599,-64075,103678,-167757,271439,-439200,

%U 710643,-1149847,1860494,-3010345,4870843,-7881192

%N Inverse binomial transform of -2 followed by A000032(n+1).

%C A simple transform of a(n) is a(n) with -a(0) instead of nonzero a(0) (or -a(0) followed by a(n+1)). Example: -1 followed by A198631(n+1)/A006519(n+2). Its inverse binomial transform is -1, 3/2, -2, 9/4, -2, 3/2, -2,... = -(-1)^n*A143074(n).

%C Difference table of -2 followed by A000032(n+1):

%C -2, 1, 3, 4, 7, 11, 18,...

%C 3, 2, 1, 3, 4, 7, 11,...

%C -1, -1, 2, 1, 3, 4, 7,...

%C 0, 3, -1, 2, 1, 3, 4,...

%C 3, -4, 3, -1, 2, 1, 3,...

%C -7, 7, -4, 3, -1, 2, 1,...

%C 14, -11, 7, -4, 3, -1, 2,...

%C etc.

%C a(n) is the first column.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,1).

%F a(n) = -2, 3, -1, followed by -(-1)^n*A206417(n).

%F a(n) = (-1)^n* (A000032(n) - 4).

%F a(n+3) = -a(n) -(-1)^n*A022112(n).

%F a(n) = -2*a(n-1) + a(n-3). - _Colin Barker_, Jun 23 2014

%F G.f.: -(5*x^2-x-2) / ((x+1)*(x^2-x-1)). - _Colin Barker_, Jun 23 2014

%o (PARI) Vec(-(5*x^2-x-2)/((x+1)*(x^2-x-1)) + O(x^100)) \\ _Colin Barker_, Jun 23 2014

%Y Cf. A000045, A000032, A000204.

%K sign,easy

%O 0,1

%A _Paul Curtz_, Jun 23 2014

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)