A110325 Row sums of number triangle related to the Jacobsthal numbers.

1, 0, -5, -14, -27, -44, -65, -90, -119, -152, -189, -230, -275, -324, -377, -434, -495, -560, -629, -702, -779, -860, -945, -1034, -1127, -1224, -1325, -1430, -1539, -1652, -1769, -1890, -2015, -2144, -2277, -2414, -2555, -2700, -2849, -3002, -3159, -3320, -3485, -3654, -3827, -4004, -4185, -4370

Offset: 0

Author

Paul Barry, Jul 20 2005

Keywords

• easy
• sign

Comments

Essentially the same sequence as A014106.

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, ...).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Cf. A005408, A110325.

Programs

Magma

[1+n-2*n^2: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

Mathematica

CoefficientList[Series[(1-3x-2x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
LinearRecurrence[{3,-3,1},{1,0,-5},50] (* Harvey P. Dale, Oct 20 2024 *)

PARI

a(n)=1+n-2*n^2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

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

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

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012

From Elmo R. Oliveira, Nov 02 2024: (Start)

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

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