On the probability of the first player reaching m first if, starting at 0, at  each round they go one step right or left each
                              with probability 1/2



                              By Shalosh B. Ekhad



Consider  a game where players take turns tossing a fair coin and at each st\
    ep either move left or right one unit

        and whoever reaches the location m first is declared the winner.

What can you say about the probability of the first player  winning the game\
    ?



           Theorem: The probabability of the first player winning is



                                 1/2 + 1/2 f(m)



               where f(m) satisfies the inhomogeneus recurrence



    2                   2                             2
(2 m  + 3 m) f(m) + (2 m  + 5 m + 2) f(m + 2) + (-12 m  - 24 m - 10) f(m + 1)

          8
     = - ----
          Pi



                       subject to the initial conditions

                              -4 + Pi           -16 + 5 Pi
                     f(1) = - -------, f(2) = - ----------
                                Pi                  Pi

                              and in Maple format



(2*m^2+3*m)*f(m)+(2*m^2+5*m+2)*f(m+2)+(-12*m^2-24*m-10)*f(m+1) = -8/Pi


                       subject to the initial conditions

f(1) = -(-4+Pi)/Pi, f(2) = -(-16+5*Pi)/Pi


Just for fun here is the sum-of-squares of the coefficient of the probabilit\
    y generating function

       for the duration until reaching, 1000, dollars for the first time

- (-894750731048126960334550414054167657495496754559952521479378919389486996\
    188198927878705222647050127632244271689222116303151848996738520313588907\
    058093986528260592422630843946157471455299439589848909686153024581224032\
    299219234759272314904792102984457906535077248129954156449458868636931019\
    336836752919389310311320437672844287231353365707839076051076388247900330\
    280087874522493139306519561203959763395876547807472262823352552748935341\
    371931133307131190608039201465392660481087213620857930580975438250432308\
    910023901386674922773938453842703986037539787689843845453496634820036208\
    652779871413775197725946139756314705826097751753971663122609302700453008\
    723440729101786585394282792772520872410635021353590695924716061934651498\
    282852733489653989925783851398697180976797247087897564826251820370148637\
    411439695876476868606232820517167582667187239228060618991402356177960417\
    926481598004145869108714908223282835891391773885353344766433712780885005\
    595227889482481772306562186281702794845073272961204414821155171429455587\
    522323426931170187919418588864846160056664001993286560923949100898476375\
    663662485623923205527522737133420228305047893926261475354868141292836596\

    707017505748059006173577024830198258944 + 284808003362792790975860939180\
    969927870152028347043838122092475984735814288365064271428408685031614561\
    814064080791317455728008540887533271112486303814831825910693790167007893\
    928767053589740859069428039349791171429952565996701330686373088319491985\
    679259859803275437231347436045002684572567586566205418573426762519194744\
    513064800010760612481427600877017148987454865210800649685229572897348679\
    677462711167397678406447909529791722090138710526718141578357411461452468\
    283770684573328484780621007242112096425616212842735132799294476924477210\
    155970476152662206590512891505757923819512480380309098594345370222851464\
    012662476080308533045943504621525784457556783558916053728488609151731706\
    386198557519533931466857154587967966231997357070134109265126996783553196\
    265526927375656512821366069314665310834824091308325215614822411266930806\
    896205235723623209871003832859408752324423810644841574963310761981285092\
    015138907143284853761675109269701928413097630520936322003218421979886586\
    788843727792216262373999791465033724849432169611531286620422018698062514\
    612062089654831065857416151236867840200568886524379879502227629807838079\
    772981720424465589080555186438465220875320645188901878190616509022274065\

    679053125 Pi)/(106693984594484749957590759035474630637397612276788389614\
    044570030446660793539144807777241258289505311777385910853317416681337764\
    422009836737695626860409648477703965265807540259441791575927747002290101\
    689004486047433234003018363597465232223312137721669492475788998276789684\
    219217668740392261231976440222751328301555544301401814506273003986039493\
    261297456873405856052770116070600603478144038208393634749025558062255637\

    84050125125 Pi)



                              and in Maple format



-1/1066939845944847499575907590354746306373976122767883896140445700304466607935\
3914480777724125828950531177738591085331741668133776442200983673769562686040964\
8477703965265807540259441791575927747002290101689004486047433234003018363597465\
2322233121377216694924757889982767896842192176687403922612319764402227513283015\
5554430140181450627300398603949326129745687340585605277011607060060347814403820\
839363474902555806225563784050125125*(-8947507310481269603345504140541676574954\
9675455995252147937891938948699618819892787870522264705012763224427168922211630\
3151848996738520313588907058093986528260592422630843946157471455299439589848909\
6861530245812240322992192347592723149047921029844579065350772481299541564494588\
6863693101933683675291938931031132043767284428723135336570783907605107638824790\
0330280087874522493139306519561203959763395876547807472262823352552748935341371\
9311333071311906080392014653926604810872136208579305809754382504323089100239013\
8667492277393845384270398603753978768984384545349663482003620865277987141377519\
7725946139756314705826097751753971663122609302700453008723440729101786585394282\
7927725208724106350213535906959247160619346514982828527334896539899257838513986\
9718097679724708789756482625182037014863741143969587647686860623282051716758266\
7187239228060618991402356177960417926481598004145869108714908223282835891391773\
8853533447664337127808850055952278894824817723065621862817027948450732729612044\
1482115517142945558752232342693117018791941858886484616005666400199328656092394\
9100898476375663662485623923205527522737133420228305047893926261475354868141292\
836596707017505748059006173577024830198258944+284808003362792790975860939180969\
9278701520283470438381220924759847358142883650642714284086850316145618140640807\
9131745572800854088753327111248630381483182591069379016700789392876705358974085\
9069428039349791171429952565996701330686373088319491985679259859803275437231347\
4360450026845725675865662054185734267625191947445130648000107606124814276008770\
1714898745486521080064968522957289734867967746271116739767840644790952979172209\
0138710526718141578357411461452468283770684573328484780621007242112096425616212\
8427351327992944769244772101559704761526622065905128915057579238195124803803090\
9859434537022285146401266247608030853304594350462152578445755678355891605372848\
8609151731706386198557519533931466857154587967966231997357070134109265126996783\
5531962655269273756565128213660693146653108348240913083252156148224112669308068\
9620523572362320987100383285940875232442381064484157496331076198128509201513890\
7143284853761675109269701928413097630520936322003218421979886586788843727792216\
2623739997914650337248494321696115312866204220186980625146120620896548310658574\
1615123686784020056888652437987950222762980783807977298172042446558908055518643\
8465220875320645188901878190616509022274065679053125*Pi)/Pi