%I #37 Apr 04 2024 10:39:38
%S 1,3,9,28,90,297,1001,3432,11933,41971,149017,533141,1919215,6942950,
%T 25215181,91858456,335449202,1227312350,4496994689,16496266812,
%U 60566602692,222524531559,817997639090,3008175954887,11066005530460,40717739034761
%N 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).
%C The sequence is constructed from a truncated version of Pascal's Triangle.
%C 1
%C 1 1
%C 1 2 1
%C 3 3 1
%C 3 6 4 1
%C 9 10 5 1
%C 9 19 15 6 1
%C 28 34 21 7 1
%C 28 62 55 28 8
%C 90 117 83 36 8
%C 90 207 200 119 44
%C 297 407 319 163 44
%C ...
%C 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.
%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) for n >= 5.
%t LinearRecurrence[{9, -28, 35, -15, 1},{1,3,9,28,90},26] (* _James C. McMahon_, Mar 12 2024 *)
%Y Cf. A007318, A211216, A224422, A221863, A122588.
%K nonn,easy
%O 0,2
%A _Peter Morris_, Feb 08 2024