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

0,2

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

Table of n, a(n) for n=0..24.

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

Formula

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

Mathematica

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

Crossrefs

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

Sequence in context: A094825 A022635 A336602 * A217274 A000588 A005285

Adjacent sequences: A370388 A370389 A370390 * A370392 A370393 A370394

Keyword

nonn,easy

Author

Peter Morris, Feb 22 2024

Status

approved