A370568 Expansion of g.f. (1-x) / (1-9*x+28*x^2-35*x^3+15*x^4-x^5).

1, 8, 44, 207, 896, 3689, 14706, 57361, 220363, 837430, 3157440, 11835916, 44176890, 164355675, 609981045, 2259680355, 8359285126, 30890694534, 114059719703, 420887785505, 1552362630016, 5723494732725, 21096366345741, 77742879583057, 286445422547405

Offset: 0

Author

Peter Morris, Feb 22 2024

Keywords

• nonn
• easy

Comments

The sequence is constructed from a truncated version of Pascal's Triangle. See A370074 for an example. a(n) arises from the Gambler's Ruin problem and represents the number of ways a gambler is ruined after starting with $8 with a maximum $11 causing retirement.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

Crossrefs

Cf. A211216, A224422, A221863, A122588, A370074, A370051, A370391.

Programs

Mathematica

LinearRecurrence[{9, -28, 35, -15, 1}, {1, 8, 44, 207, 896}, 25] (* Paolo Xausa, Jun 09 2024 *)

Formula

a(n) = 9*a(n-1)-28*a(n-2)+35*a(n-3)-15*a(n-4)+a(n-5) for n>=5.