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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).

A370074 Expansion of (1 - 2x) (1 - 4x + 2x^2) / (1 - 9x + 28x^2 - 35x^3 + 15x^4 - x^5).

Original entry on oeis.org

1, 3, 9, 28, 90, 297, 1001, 3432, 11933, 41971, 149017, 533141, 1919215, 6942950, 25215181, 91858456, 335449202, 1227312350, 4496994689, 16496266812, 60566602692, 222524531559, 817997639090, 3008175954887, 11066005530460, 40717739034761

Offset: 0

Peter Morris, Feb 08 2024

The sequence is constructed from a truncated version of Pascal's Triangle.
           1
       1       1
   1       2       1
       3       3       1
   3       6       4       1
       9      10       5       1
   9      19      15       6       1
      28      34      21       7       1
  28      62      55      28       8
      90     117      83      36       8
  90     207     200     119      44
     297     407     319     163      44
         ...
After truncation the sequence appears as the left vertical column. The right column sequence can be found in A370568. a(n) arises from the Gambler's Ruin problem and it represents the number of ways a gambler is ruined in the Gambler's Ruin problem starting with $3 and with a maximum $11 causing retirement.

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

Cf. A007318, A211216, A224422, A221863, A122588.

LinearRecurrence[{9, -28, 35, -15, 1},{1,3,9,28,90},26] (* 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) for n >= 5.

A370051 Expansion of (1-5x+6x^2-x^3)/(1-9x+28x^2-35x^3+15x^4-x^5).

Original entry on oeis.org

1, 4, 14, 48, 165, 572, 2002, 7071, 25176, 90251, 325358, 1178291, 4282811, 15612092, 57040186, 208772476, 765186422, 2807556411, 10309833845, 37883902913, 139275229088, 512223805060, 1884404481767, 6934058102453, 25519786076294

Offset: 0

Peter Morris, Feb 08 2024

In Pascal's triangle, subtract the 6th column to the left of the central column from the 2nd column.

			a(0) = binomial(2,0);
a(1) = binomial(4,1);
a(2) = binomial(6,2) - binomial(6,0);
a(3) = binomial(8,3) - binomial(8,1);
a(4) = binomial(10,4) - binomial(10,2).

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

Cf. A007318, A211216, A224422, A221863, A122588.

LinearRecurrence[{9, -28, 35, -15, 1}, {1, 4, 14, 48, 165}, 30] (* Paolo Xausa, Feb 20 2024 *)

a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5); a(0) = 1, a(1) = 4, a(2) = 14, a(3) = 48, a(4) = 165.

A336602 a(n) = 8a(n-1) - 21a(n-2) + 20a(n-3) - 5a(n-4), with initial terms a(0)=1, a(1)=7, a(2)=35, a(3)=154.

Original entry on oeis.org

1, 7, 35, 154, 632, 2487, 9529, 35875, 133471, 492538, 1807268, 6604891, 24069905, 87539199, 317907067, 1153307002, 4180842064, 15147734815, 54860799881, 198634274203, 719047882103, 2602540622106, 9418700937340, 34084040705539, 123335178991777, 446277892754167, 1614771692630099

Offset: 0

Peter Morris, Dec 20 2020

Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Extension of patterns illustrated in A001519, A033191, A033190, A094667, A030191, A094788.

G.f.: ( 1-x+x^3 ) / ( (5*x^2-5*x+1)*(x^2-3*x+1) ). - R. J. Mathar, May 05 2023

Offset corrected by Jon E. Schoenfield, Feb 05 2021

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User: Peter Morris

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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A370074 Expansion of (1 - 2*x) * (1 - 4*x + 2*x^2) / (1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5).

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A370051 Expansion of (1-5*x+6*x^2-x^3)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).

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A336602 a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), with initial terms a(0)=1, a(1)=7, a(2)=35, a(3)=154.

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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).

A370074 Expansion of (1 - 2x) (1 - 4x + 2x^2) / (1 - 9x + 28x^2 - 35x^3 + 15x^4 - x^5).

A370051 Expansion of (1-5x+6x^2-x^3)/(1-9x+28x^2-35x^3+15x^4-x^5).

A336602 a(n) = 8a(n-1) - 21a(n-2) + 20a(n-3) - 5a(n-4), with initial terms a(0)=1, a(1)=7, a(2)=35, a(3)=154.