A110325 - OEIS

%I #27 Nov 02 2024 14:58:15

%S 1,0,-5,-14,-27,-44,-65,-90,-119,-152,-189,-230,-275,-324,-377,-434,

%T -495,-560,-629,-702,-779,-860,-945,-1034,-1127,-1224,-1325,-1430,

%U -1539,-1652,-1769,-1890,-2015,-2144,-2277,-2414,-2555,-2700,-2849,-3002,-3159,-3320,-3485,-3654,-3827,-4004,-4185,-4370

%N Row sums of number triangle related to the Jacobsthal numbers.

%C Essentially the same sequence as A014106.

%C Rows sums of A110324. Results from a general construction: the row sums of the inverse of the number triangle whose columns have e.g.f. (x^k/k!)/(1 - a*x - b*x^2), g.f. (1 - (a+2)*x - (2*b-a-1)*x^2)/(1-x)^3, and general term 1 + (b-a)*n - b*n^2. This is the binomial transform of (1, -a, -2b, 0, 0, 0, ...).

%H Vincenzo Librandi, <a href="/A110325/b110325.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F G.f.: (1 - 3*x - 2*x^2)/(1-x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jul 08 2012

%F From _Elmo R. Oliveira_, Nov 02 2024: (Start)

%F E.g.f.: exp(x)*(1 - x - 2*x^2).

%F a(n) = -A005408(n)*A110325(n). (End)

%t CoefficientList[Series[(1-3x-2x^2)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 08 2012 *)

%t LinearRecurrence[{3,-3,1},{1,0,-5},50] (* _Harvey P. Dale_, Oct 20 2024 *)

%o (Magma) [1+n-2*n^2: n in [0..50]]; // _Vincenzo Librandi_, Jul 08 2012

%o (PARI) a(n)=1+n-2*n^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A014106 (essentially the same sequence), A110324.

%Y Cf. A005408, A110325.

%K easy,sign

%O 0,3

%A _Paul Barry_, Jul 20 2005