A Linear Recurrence With Polynomial Coefficients Satisfied by The Inegral of\ the, 2, -th power of Certain C-finite Polynomial Sequence By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence P[n](x) = P[n - 1](x) + x P[n - 2](x) Subject to the initial conditions P[0] = 1, P[1] = 1 Equivalently, in terms of generating function infinity ----- \ n 1 ) P[n](x) t = ------------- / 2 ----- -t x - t + 1 n = 0 1 n Define the Umbra, U, to be equal to, -----, when applied to , x , n + 1 and extended linearly Let's define the sequence of numbers 2 s(n), to be the action of U applied to, P[n](x) Then s(n) satisfies the following linear recurrence equation with polynomial\ coefficients 2 (2 n + 3) (n + 2) (4640726 n + 13959505 n - 32195751) s(n) -1/4 ----------------------------------------------------------- + 1/4 (2 n + 13) (n + 7) %1 4 3 2 (66696752 n + 443766648 n + 219969574 n - 2821002751 n - 4251164619) s(n + 1)/((2 n + 13) (n + 7) %1) - 1/4 4 3 2 (123463540 n + 793061828 n - 1012483739 n - 14116590370 n - 22009535668) s(n + 2)/((2 n + 13) (n + 7) %1) - 1/4 4 3 2 (7125300 n + 326902512 n + 4169609581 n + 19319181673 n + 29826152632) s(n + 3)/((2 n + 13) (n + 7) %1) + 1/4 ( 4 3 2 125838640 n + 1151331512 n + 1301115838 n - 11172916699 n - 22952362818) s(n + 4)/((2 n + 13) (n + 7) %1) - 1/4 4 3 2 (59571452 n + 369400544 n - 3371322551 n - 29907213998 n - 55438569843) s(n + 5)/((2 n + 13) (n + 7) %1) + s(n + 6) = 0 2 %1 := 863294 n - 11654469 n - 49818801 Subject to the initial condition 281 302 9487 s(1) = 1, s(2) = 7/3, s(3) = 13/3, s(4) = ---, s(5) = ---, s(6) = ---- 30 15 210 Just for fun, using this recurrence we get that s(1000) = 248984253891696722549173713296461719558136921264843480369154408816\ 399309679593518023939130240047544242587111473586716310711318735245170732\ 292733605923040871716914971055320768093713921448699799778256510924578786\ 829472211010041007676343517695070703432204246176404477421849107080372518\ 204482012527995256560863941044185267918602912538394129847304812338344996\ 865857119784244743778246237046387360125875965840244074004759366004660939\ 903181212971091934227256750642666180464823555713353281253928747728184507\ 777600307976073663409371057078448759631588014528228587783605136038723279\ 115133289966299389713658796511842343422325372534198892438278315532980707\ 993199619722900942181454647958270894612378217584932088658572290940365489\ 635652384451963448439462191454272281368294026842571376813009276287700088\ / 101884721796524711830531897093321463988972581585311941916789 / 2784712\ / 997916051973893118810825887859636077680424176968926563277794657846711371\ 679482985996841356088637389772273271584575382915651414456738853855861056\ 691825268073493437576800771430760131714196759771654083017085838007407478\ 779289893842561028446794535573753618092855024210758121581154124238018154\ 585089813809668670599706266587358613725404035630774119863624395892842977\ 3000294426757507795593972390738669495670654248721476370826576250 This ends this article, that took, 1.154, seconds to generate. ---------------------------------------------------- IntCv([[1,1,1,x],[1,x,x,x^2]],x,1,n,N,1/(n+1),20,1000); ----------------------------------------------------------------------------\ ---------------- A Linear Recurrence With Polynomial Coefficients Satisfied by The Inegral of\ the, 1, -th power of Certain C-finite Polynomial Sequence By Shalosh B. Ekhad Proposition: Let , P[n](x), be the sequence of polynomial definied by the following recurrence 2 P[n](x) = P[n - 1](x) + x P[n - 2](x) + x P[n - 3](x) + x P[n - 4](x) Subject to the initial conditions P[0] = 1, P[1] = 1, P[2] = 1, P[3] = x Equivalently, in terms of generating function infinity ----- 2 3 \ n 1 - x t + (-x - 1) t ) P[n](x) t = ---------------------------- / 4 2 3 2 ----- -t x - t x - t x - t + 1 n = 0 1 n Define the Umbra, U, to be equal to, -----, when applied to , x , n + 1 and extended linearly Let's define the sequence of numbers s(n), to be the action of U applied to, P[n](x) Then s(n) satisfies the following linear recurrence equation with polynomial\ coefficients 4 -1/5 n (n + 1) (367877617509599685914493538627337782773 n 3 + 6789127038228067682522303232203272403835 n 2 + 45531480225792661448807233442908104810589 n + 133318662482612598310843093516067719604656 n + 143837670075418983257871382093736262840519) s(n)/((n + 10) (n + 9) %1) 5 + 1/5 (n + 1) (890261852506892475358895300997141690267 n 4 + 18017703602213274416677126460364608620362 n 3 + 133841243125549990985765068192710580552980 n 2 + 445927790649910028775347219634452212682715 n + 628517900701777300921438805724373917272994 n + 262461122277735611280524705531409488067218) s(n + 1)/((n + 10) (n + 9) 6 %1) - 1/5 (2148401322523384636632284140621621163307 n 5 + 46017704035172586731397643818293718622854 n 4 + 375549174027735149444047241796279257729356 n 3 + 1474778123930301930741947010814777127450300 n 2 + 2869756844981472628970440879347776443310643 n + 2562331926120896207563677831456338486539426 n + 835539273324742028411876378046802146252170) s(n + 2)/((n + 10) (n + 9) 6 %1) + 1/5 (772533087438465517649541118712330623605 n 5 + 21689554716092721281504629209193419738753 n 4 + 213995315302143052587412040588545978354802 n 3 + 962890291908303241920960695332439437973809 n 2 + 2111099543373522166243939399266865729090044 n + 2202234471238586162411483491848940697915449 n + 854164252036416452845478477243637805976214) s(n + 3)/((n + 10) (n + 9) 6 %1) + 1/5 (735755235019199371828987077254675565546 n 5 + 18110514251639758553727883110219409267608 n 4 + 179455429906065608595224854336653365730923 n 3 + 902390067406990841674233359662417199538062 n 2 + 2342981077978219742693964959084468344017159 n + 2846276908771800233980932858228852561768500 n + 1097916759266300602919789796650085239562956) s(n + 4)/((n + 10) (n + 9) 6 %1) - 1/5 (2148401322523384636632284140621621163307 n 5 + 53080934472693513055414129135128446611659 n 4 + 511051821340685904363396905862602613821746 n 3 + 2426841010097291570469302552245919429221377 n 2 + 5839319995928641734678143741451192954049946 n + 6468863924743270301655949411086809851157783 n + 2519289911035710692173041070246459678811114) s(n + 5)/((n + 10) (n + 9) 6 %1) + 1/5 (5187064497553661748623463582240384016881 n 5 + 133435377367874506154222400287447351946336 n 4 + 1338914466329906653348653779485400029204409 n 3 + 6647649226601448814814844251844312768022688 n 2 + 16992868168968898719385581544564872959470599 n + 20749988974601513167668069531308353834463963 n + 9023750283047614063532822083157279171207000) s(n + 6)/((n + 10) (n + 9) 5 %1) - 1/5 (3693467497400315671931366378046282410517 n 4 + 72224393302792209241844013227531280578874 n 3 + 517347441374648448607592143260767001915866 n 2 + 1651534316650381418705927998470008822925508 n + 2267340574005395809023567343537438255455848 n + 1030887720611589708690992688085202573062272) s(n + 7)/((n + 10) %1) + s(n + 8) = 0 4 %1 := 154506617487693103529908223742466124721 n 3 + 2004121498373315540974542457536260875215 n 2 + 8364757802215905519923679231318585166843 n + 12687598270015648933216178839716251025773 n + 5906094262223422381677701332105521313238 Subject to the initial condition 59 s(1) = 1, s(2) = 1, s(3) = 1/2, s(4) = 11/6, s(5) = 3, s(6) = --, s(7) = 17/2, 12 226 s(8) = --- 15 Just for fun, using this recurrence we get that s(1000) = 202553239580995654879095892718245263301469097677728719157189760851\ 136816167679403873694376401861484421609910976430710995183180839487418872\ 812070452080784199339361858249060073597905359178394784440000300374884163\ 714886525086017183084325813798575473046415181459810975919968972117355390\ 923918637576710627761670691563211552204415082480747825727048779533917800\ 463028045681432000490651802329652889199827946008676498495543664087731165\ / 3099110113014947813209677076202938745099464031905510998352031235853247 / / 244107663630146473998813033760098319600577736397324464374726494132574433\ 959283116336125586840953661251435898432318131105933786665712173239657075\ 7613027357375997780740624450832125250548969734546060400533484613856000 This ends this article, that took, 3.956, seconds to generate. ----------------------------------------------------