A229463 Expansion of g.f. 1/((1-x)^2*(1-26*x)).

1, 28, 731, 19010, 494265, 12850896, 334123303, 8687205886, 225867353045, 5872551179180, 152686330658691, 3969844597125978, 103215959525275441, 2683614947657161480, 69773988639086198495, 1814123704616241160886, 47167216320022270183053, 1226347624320579024759396

Offset: 0

Author

Yahia Kahloune, Sep 24 2013

Keywords

• nonn
• easy

Comments

This sequence was chosen to illustrate a method of solution.

Examples

			a(3) = (26^5 - 25*3 - 51)/625 = 19010.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (28,-53,26).

Crossrefs

Cf. A000295, A000340, A014825, A014827, A014936.

Programs

PARI

my(x='x+O('x^18)); Vec(1/((1-26*x)*(1-x)^2)) \\ Elmo R. Oliveira, May 24 2025

Formula

a(n) = (26^(n+2) - 25*n - 51)/625.

In general, for the expansion of 1/((1-s*x)^2*(1-r*x)) with r>s>=1 we have the formula: a(n) = (r^(n+2)- s^(n+1)*((r-s)*n +(2*r-s)))/(r-s)^2.

From Elmo R. Oliveira, May 24 2025: (Start)

E.g.f.: exp(x)*(-51 - 25*x + 676*exp(25*x))/625.

a(n) = 28*a(n-1) - 53*a(n-2) + 26*a(n-3). (End)