A317509 Coefficients in expansion of 1/(1 + x - 2*x^5).

1, -1, 1, -1, 1, 1, -3, 5, -7, 9, -7, 1, 9, -23, 41, -55, 57, -39, -7, 89, -199, 313, -391, 377, -199, -199, 825, -1607, 2361, -2759, 2361, -711, -2503, 7225, -12743, 17465, -18887, 13881, 569, -26055, 60985, -98759, 126521

Offset: 0

Author

Shara Lalo, Sep 04 2018

Keywords

• sign
• easy

Comments

Coefficients in expansion of 1/(1 + x - 2*x^5) are given by the sum of numbers along "fourth Layer" skew diagonals pointing top-left in triangle A065109 ((2-x)^n) and by the sum of numbers along "fourth Layer" skew diagonals pointing top-right in triangle A303872 ((-1+2*x)^n), see links.

References

• Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Links

• Shara Lalo, Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (2 - x)^n
• Shara Lalo, Fourth layer skew diagonals in center-justified triangle of coefficients in expansion of (-1 + 2x)^n
• Index entries for linear recurrences with constant coefficients, signature (-1,0,0,0,2).

Crossrefs

Cf. A065109, A303872.

Programs

Mathematica

CoefficientList[Series[1/(1 + x - 2 x^5), {x, 0, 42}], x]
a[0] = 1; a[n_] := a[n] = If[n < 0, 0, - a[n - 1] + 2 * a[n - 5]]; Table[a[n], {n, 0, 42}] // Flatten
LinearRecurrence[{-1,0,0,0,2}, {1,-1,1,-1,1}, 43]

PARI

my(x='x+O('x^99)); Vec(1/(1+x-2*x^5)) \\ Altug Alkan, Sep 04 2018

Formula

a(0)=1, a(n) = -1 * a(n-1) + 2 * a(n-5) for n >= 0; a(n)=0 for n < 0.