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).

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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..25.

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) for n >= 5.

Mathematica

LinearRecurrence[{9, -28, 35, -15, 1}, {1, 3, 9, 28, 90}, 26] (* James C. McMahon, Mar 12 2024 *)

Crossrefs

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

Sequence in context: A094803 A094826 A033190 * A071724 A000245 A143739

Adjacent sequences: A370071 A370072 A370073 * A370075 A370076 A370077

Keyword

nonn,easy

Author

Peter Morris, Feb 08 2024

Status

approved