A370391 Expansion of (1 - 2*x)/(1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5).

1, 7, 35, 154, 636, 2533, 9861, 37810, 143451, 540155, 2022735, 7543771, 28048829, 104050724, 385320419, 1425038684, 5264963100, 19437087382, 71715418017, 264483764116, 975070823122, 3593840295815, 13243217176106, 48793364067681, 179753027448972

Offset: 0

Author

Peter Morris, Feb 22 2024

Keywords

• nonn
• easy

Comments

The sequence is constructed by a truncated version of Pascal's Triangle.

1

1 1

1 2 1

1 3 3 1

1 4 6 4

1 5 10 10 4

1 6 15 20 14

7 21 35 34 14

7 28 56 69 48

35 84 125 117 48

35 119 209 242 165

...

After truncation the sequence appears as the left vertical column. The right column sequence can be in A370051.

a(n) arises from the Gambler's Ruin problem and represents the number of ways a gambler is ruined after starting with $7 with a maximum $11 causing retirement.

Links

• Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

Crossrefs

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

Programs

Mathematica

LinearRecurrence[{9, -28, 35, -15, 1}, {1, 7,35,154,636}, 25] (* James C. McMahon, Mar 12 2024 *)

Formula

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