A073358 -id:A073358 - cp's OEIS frontend

A124342 Expansion of (1+x)/(1+2x-2x^3).

1, -1, 2, -2, 2, 0, -4, 12, -24, 40, -56, 64, -48, -16, 160, -416, 800, -1280, 1728, -1856, 1152, 1152, -6016, 14336, -26368, 40704, -52736, 52736, -24064, -57344, 220160, -488448, 862208, -1284096, 1591296, -1458176, 348160

Offset: 0

Paul Barry, Oct 26 2006

Diagonal sums of A124341. Binomial transform has g.f. (1-x)/(1-x-x^2-x^3).

Index entries for linear recurrences with constant coefficients, signature (-2,0,2).

a(n)=sum{k=0..floor(n/2), sum{j=0..n-k, (-1)^(n-k-j)*C(n-k,j)*C(k,j-k)}}
a(n) = (-1)^(n+1)*A073358(n+1). - R. J. Mathar, Feb 04 2014
a(n) = A077988(n-1)+A077988(n). - R. J. Mathar, Jan 25 2016

A144413 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4).

Original entry on oeis.org

0, 1, -1, 1, 0, -3, 10, -24, 49, -89, 145, -208, 245, -174, -176, 1121, -3185, 7137, -13920, 24301, -37926, 51256, -53615, 20407, 97265, -386224, 984549, -2083934, 3896480, -6537023, 9734175, -12231999, 10690624, 2126301, -39992150, 126414472, -297132815, 598577351, -1075051951, 1730868336, -2443923755

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (-3,-2,1).

Cf. A000073, A000931, A073358.

I:=[0,1,1]; [n le 3 select I[n] else -3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..51]]; // G. C. Greubel, Mar 27 2021

(* First program *)
b[n_]:= b[n]= If[n==0, 0, If[n<3, 1, b[n-2] +b[n-3]]];
a[n_]:= Sum[(-1)^m*Binomial[n, m]*b[n-m], {m,0,n}];
Table[a[n], {n, 0, 50}]
(* Second program *)
LinearRecurrence[{-3,-2,1}, {0,1,-1}, 51] (* G. C. Greubel, Mar 27 2021 *)

@CachedFunction
def A000931(n): return 1 if n==0 else sum( binomial(k, n-2*k-3) for k in (0..floor((n-3)/2)))
def A144413(n): return sum( (-1)^k*binomial(n, k)*A000931(n-k+4) for k in (0..n))
[A144413(n) for n in (0..50)] # G. C. Greubel, Mar 27 2021

From R. J. Mathar, Jan 21 2009: (Start)
a(n) = -3*a(n-1) - 2*a(n-2) + a(n-3).
G.f.: x*(1 +2*x)/(1 +3*x +2*x^2 -x^3). (End)

Terms a(30) onward added and edited by G. C. Greubel, Mar 27 2021

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A124342 Expansion of (1+x)/(1+2x-2x^3).

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A144413 a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * A000931(n-k+4).

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