A370051 -id:A370051 - cp's OEIS frontend

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

Peter Morris, Feb 22 2024

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.

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).

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

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

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

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

Original entry on oeis.org

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

Peter Morris, Feb 22 2024

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.

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

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

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

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

cp's OEIS Frontend

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

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

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cp's OEIS Frontend

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

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

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

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