OFFSET
0,2
COMMENTS
The coefficients in the expansion of 1/(1 + 2*x - 3*x^3) are given by the sequence generated by the row sums in triangle A317503.
REFERENCES
Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 396, 397.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 - 2x)^n
Zagros Lalo, Second layer skew diagonals in center-justified triangle of coefficients in expansion of (-2 + 3x)^n
Index entries for linear recurrences with constant coefficients, signature (-2,0,3).
FORMULA
a(0)=1, a(n) = -2*a(n-1) + 3*a(n-3) for n = 0,1...; a(n)=0 for n < 0.
a(n) = (2^(-n)*(2^n + (-3-i*sqrt(3))^n*(3-2*i*sqrt(3)) + (-3+i*sqrt(3))^n*(3+2*i*sqrt(3)))) / 7 where i=sqrt(-1). - Colin Barker, Aug 02 2018
MAPLE
seq(coeff(series(1/(1+2*x-3*x^3), x, n+1), x, n), n=0..40); # Muniru A Asiru, Aug 01 2018
MATHEMATICA
CoefficientList[Series[1/(1 + 2 x - 3 x^3), {x, 0, 40}], x].
a[0] = 1; a[n_] := a[n] = If[n < 0, 0, -2 * a[n - 1] + 3 * a[n - 3]]; Table[a[n], {n, 0, 40}] // Flatten.
LinearRecurrence[{-2, 0, 3}, {1, -2, 4}, 41].
PROG
(GAP) a:=[1, -2, 4];; for n in [4..40] do a[n]:=-2*a[n-1]+3*a[n-3]; od; a; # Muniru A Asiru, Aug 01 2018
(PARI) Vec(1 / ((1 - x)*(1 + 3*x + 3*x^2)) + O(x^40)) \\ Colin Barker, Aug 02 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Zagros Lalo, Jul 31 2018
STATUS
approved