A318777 Coefficients in expansion of 1/(1 - x - 2*x^5).

1, 1, 1, 1, 1, 3, 5, 7, 9, 11, 17, 27, 41, 59, 81, 115, 169, 251, 369, 531, 761, 1099, 1601, 2339, 3401, 4923, 7121, 10323, 15001, 21803, 31649, 45891, 66537, 96539, 140145, 203443, 295225, 428299, 621377, 901667, 1308553, 1899003, 2755601, 3998355, 5801689, 8418795

Offset: 0

Author

Zagros Lalo, Sep 25 2018

Keywords

• nonn
• easy

Comments

The coefficients in the expansion of 1/(1 - x - 2*x^5) are given by the sequence generated by the row sums in triangle A318775.

Coefficients in expansion of 1/(1 - x - 2*x^5) are given by the sum of numbers along "fourth Layer" skew diagonals pointing top-right in triangle A013609 ((1+2x)^n) and by the sum of numbers along "fourth Layer" skew diagonals pointing top-left in triangle A038207 ((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

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

Crossrefs

Cf. A318775, A318776.

Cf. A013609, A038207.

Essentially a duplicate of A143447.

Programs

GAP

a:=[1,1,1,1,1,3];; for n in [7..50] do a[n]:=a[n-1]+2*a[n-5]; od; a; # Muniru A Asiru, Sep 26 2018

Maple

seq(coeff(series((1-x-2*x^5)^(-1),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Sep 26 2018

Mathematica

a[0] = 1; a[n_] := a[n] = If[n < 0, 0, a[n - 1] + 2 * a[n - 5]];Table[a[n], {n, 0, 45}] // Flatten
LinearRecurrence[{1, 0, 0, 0, 2}, {1, 1, 1, 1, 1}, 46]
CoefficientList[Series[1/(1 - x - 2 x^5), {x, 0, 45}], x]

Formula

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

G.f.: -1/(2*x^5 + x - 1). - Chai Wah Wu, Aug 03 2020