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

%I #43 Apr 04 2024 10:40:13

%S 1,7,35,154,636,2533,9861,37810,143451,540155,2022735,7543771,

%T 28048829,104050724,385320419,1425038684,5264963100,19437087382,

%U 71715418017,264483764116,975070823122,3593840295815,13243217176106,48793364067681,179753027448972

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

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

%C 1

%C 1 1

%C 1 2 1

%C 1 3 3 1

%C 1 4 6 4

%C 1 5 10 10 4

%C 1 6 15 20 14

%C 7 21 35 34 14

%C 7 28 56 69 48

%C 35 84 125 117 48

%C 35 119 209 242 165

%C ...

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

%C 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.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-28,35,-15,1).

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

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

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

%K nonn,easy

%O 0,2

%A _Peter Morris_, Feb 22 2024