A124750 - OEIS

%I #15 Jan 13 2022 02:26:55

%S 1,1,1,1,0,0,0,-1,0,0,-1,1,0,-1,2,-1,-1,3,-3,0,4,-6,3,4,-10,9,1,-14,

%T 19,-8,-15,33,-27,-7,48,-60,20,55,-108,80,35,-163,188,-45,-198,351,

%U -233,-153,549,-584,80,702,-1133,664,622,-1835,1797,-42,-2457,3632,-1839

%N Expansion of (1 + x + x^2)/(1 - x^3 + x^4).

%C Row sums of number triangle A124749.

%C Let A(n) denote the n X n matrix with 1's along and everywhere above the main diagonal, 1's along the sub-sub-subdiagonal, and 0's everywhere else; for n>3, a(n) equals (-1)^(n+1) times the sum of the coefficients of the characteristic polynomial of A(n-3) (see Mathematica code below). - _John M. Campbell_, Mar 10 2012

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,-1).

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

%F a(n) = Sum_{k=0..n} binomial(floor(k/3), n-k) * (-1)^(n-k).

%F a(n) = a(n-3) - a(n-4). - _Wesley Ivan Hurt_, May 02 2021

%t A[n_] := Array[Sum[KroneckerDelta[#1, #2 - j], {j, 0, n}] + KroneckerDelta[#1, #2 + 3] &, {n, n}]; Table[(-1)^(r + 1)*Total[CoefficientList[CharacteristicPolynomial[A[r - 3], x],  x]], {r, 4, 60}] (* _John M. Campbell_, Mar 10 2012 *)

%t CoefficientList[Series[(1+x+x^2)/(1-x^3+x^4),{x,0,70}],x] (* or *) LinearRecurrence[{0,0,1,-1},{1,1,1,1},70] (* _Harvey P. Dale_, Jun 06 2018 *)

%Y Cf. A124749.

%K easy,sign

%O 0,15

%A _Paul Barry_, Nov 06 2006