A110331 Row sums of a number triangle related to the Pell numbers.

1, -1, -5, -11, -19, -29, -41, -55, -71, -89, -109, -131, -155, -181, -209, -239, -271, -305, -341, -379, -419, -461, -505, -551, -599, -649, -701, -755, -811, -869, -929, -991, -1055, -1121, -1189, -1259, -1331, -1405, -1481, -1559, -1639, -1721, -1805, -1891, -1979, -2069, -2161, -2255, -2351, -2449

Offset: 0

Author

Paul Barry, Jul 20 2005

Keywords

• easy
• sign

Comments

Row sums of A110330. 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) have g.f. (1-(a+2)x-(2b-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,...).

Hankel transform of A007054(n)-2*0^n. - Paul Barry, Jul 20 2008

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. A028387 (absolute values). A165900 is another version.

Programs

Magma

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

Mathematica

CoefficientList[Series[(1-4x+x^2)/(1-x)^3,{x,0,50}],x]  (* Vincenzo Librandi, Jul 08 2012 *)
LinearRecurrence[{3,-3,1},{1,-1,-5},60] (* Harvey P. Dale, Mar 22 2022 *)

PARI

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

Formula

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

G.f.: (1-4*x+x^2)/(1-x)^3.

a(n) = binomial(n+2, 2) - 4*binomial(n+1, 2) + binomial(n, 2).

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

E.g.f.: exp(x)*(1-2*x-x^2). - Tom Copeland, Dec 02 2013

a(n) = -A165900(n+1) (= -A028387(n-1) for n > 0). - M. F. Hasler, Mar 01 2014