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A110331 Row sums of a number triangle related to the Pell numbers. 8

%I

%S 1,-1,-5,-11,-19,-29,-41,-55,-71,-89,-109,-131,-155,-181,-209,-239,

%T -271,-305,-341,-379,-419,-461,-505,-551,-599,-649,-701,-755,-811,

%U -869,-929,-991,-1055,-1121,-1189,-1259,-1331,-1405,-1481,-1559,-1639,-1721,-1805,-1891,-1979,-2069,-2161,-2255,-2351,-2449

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

%C 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,...).

%C Hankel transform of A007054(n)-2*0^n. - _Paul Barry_, Jul 20 2008

%H Vincenzo Librandi, <a href="/A110331/b110331.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-n^2.

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

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

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

%F E.g.f.: exp(x)*(1-2*x-x^2). - _Tom Copeland_, Dec 02 2013

%F a(n) = -A165900(n+1) (= -A028387(n-1) for n > 0). - _M. F. Hasler_, Mar 01 2014

%t CoefficientList[Series[(1-4x+x^2)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 08 2012 *)

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

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

%Y Cf. A028387 (absolute values). A165900 is another version.

%K easy,sign

%O 0,3

%A _Paul Barry_, Jul 20 2005

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Last modified October 1 12:43 EDT 2020. Contains 337443 sequences. (Running on oeis4.)