%I #26 Jun 13 2015 00:53:51
%S 1,26,676,17576,456972,11881168,308907672,8031529376,208817941280,
%T 5429219088800,141158464323104,3670088041052160,95421456259562432,
%U 2480936209934965120,64503778490067388160,1677083603695215199744,43603793136187040353536,1133688727070116383116288,29475649649828842801150464,766360202350076625301264384
%N Expansion of (1+2*x^2)/(1-26*x+2*x^2-52*x^3+4*x^4).
%C From the Noonan and Zeilberger link below, a(n) is the number of words in the 26-letter English alphabet {A,B,C,...,X,Y,Z} that do not contain any of the "bad" words: PIPI or CACA or PICA or CAPI. The expected wait time to see an occurrence of one of any of these four words is G(1/26) = 114582. The expected wait time to see all four of these words is 979223595402/1028195 (approximately 952371). - _Geoffrey Critzer_, May 17 2014
%H J. Noonan and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9806036">The Goulden-Jackson cluster method: extensions, applications and implementations</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (26,-2,52,-4).
%F G.f.: (1+2*x^2)/(1-26*x+2*x^2-52*x^3+4*x^4).
%F a(0)=1, a(1)=26, a(2)=676, a(3)=17576, a(n)=26*a(n-1)-2*a(n-2)+ 52*a(n-3)- 4*a(n-4). - _Harvey P. Dale_, Oct 04 2014
%t sol=Solve[{A==-z^4-z^2A-z^2D,B==-z^4-z^2B-z^2C,C==-z^4-z^2A-z^2D,D==-z^4-z^2B-z^2C},{A,B,C,D}];nn=20;CoefficientList[Series[1/(1-26z-A-B-C-D)/.sol,{z,0,nn}],z] (* _Geoffrey Critzer_, May 17 2014 *)
%t LinearRecurrence[{26,-2,52,-4},{1,26,676,17576},30] (* _Harvey P. Dale_, Oct 04 2014 *)
%Y Cf. A188696.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Apr 08 2011