A122584 Expansion of x*(1+x)*(1-2*x)/(1 - 2*x - x^2 + 2*x^3 - x^4).

1, 1, 1, 1, 2, 4, 9, 19, 41, 87, 186, 396, 845, 1801, 3841, 8189, 17462, 37232, 79389, 169275, 360937, 769603, 1640982, 3498968, 7460649, 15907905, 33919505, 72324585, 154213514, 328820508, 701124865, 1494967795, 3187632953

Offset: 1

Author

Roger L. Bagula, Sep 19 2006

Keywords

• nonn
• easy

References

• A. Messiah, Quantum mechanics, vol. 2, pp. 608-609, eq.(XIV.57), North Holland, 1969.

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..3000
• Miklos Bona and Rebecca Smith, Pattern avoidance in permutations and their squares, arXiv:1901.00026 [math.CO], 2018. See p. 6.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,1).

Crossrefs

Cf. A122581, A122582, A122583.

Programs

Magma

[n le 5 select 1 else 2*Self(n-1) +Self(n-2) -2*Self(n-3) +Self(n-4): n in [1..50]]; // G. C. Greubel, Nov 28 2021

Maple

seq(coeff(series((x*(1+x)*(2*x-1))/(x^4-2*x^3+x^2+2*x-1),x,n+1), x, n), n = 1 .. 40); # Muniru A Asiru, Jan 03 2019

Mathematica

a[n_]:= a[n]= If[n<4, 1, 2*a[n-1] +a[n-2] -2*a[n-3] +a[n-4]];
Table[a[n], {n, 50}] (* modified by G. C. Greubel, Nov 28 2021 *)

PARI

Vec(x*(1+x)*(2*x-1)/(-1+2*x+x^2-2*x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

Sage

@CachedFunction # a=A122584
def a(n): return 1 if (n<5) else 2*a(n-1) +a(n-2) -2*a(n-3) +a(n-4)
[a(n) for n in (1..50)] # G. C. Greubel, Nov 28 2021

Formula

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

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) + a(n-4).