{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013611 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-1","-26","-248","-1068","-1920","-448","1536"],["0","-12","-220","-1368","-3328","-1920","2048"],["0","0","-44","-504","-1824","-1824","896"],["0","0","0","-48","-352","-608","128"],["0","0","0","0","-16","-64","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-3","-78","-744","-3204","-5760","-1344","4608"],["0","0","-36","-660","-4104","-9984","-5760","6144"],["0","0","0","-132","-1512","-5472","-5472","2688"],["0","0","0","0","-144","-1056","-1824","384"],["0","0","0","0","0","-48","-192","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-1","-26","-248","-1068","-1920","-448","1536"],["0","0","0","-12","-220","-1368","-3328","-1920","2048"],["0","0","0","0","-44","-504","-1824","-1824","896"],["0","0","0","0","0","-48","-352","-608","128"],["0","0","0","0","0","0","-16","-64","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-2","-36","-224","-568","-384","640","1024"],["0","-24","-304","-1280","-1888","0","1536"],["0","0","-88","-688","-1632","-800","768"],["0","0","0","-96","-448","-448","128"],["0","0","0","0","-32","-64","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-6","-108","-672","-1704","-1152","1920","3072"],["0","0","-72","-912","-3840","-5664","0","4608"],["0","0","0","-264","-2064","-4896","-2400","2304"],["0","0","0","0","-288","-1344","-1344","384"],["0","0","0","0","0","-96","-192","0"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","-2","-36","-224","-568","-384","640","1024"],["0","0","0","-24","-304","-1280","-1888","0","1536"],["0","0","0","0","-88","-688","-1632","-800","768"],["0","0","0","0","0","-96","-448","-448","128"],["0","0","0","0","0","0","-32","-64","0"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["1","10","32","28","-32","-64"],["0","12","84","168","0","-160"],["0","0","44","184","128","-128"],["0","0","0","48","96","-32"],["0","0","0","0","16","0"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","3","30","96","84","-96","-192"],["0","0","36","252","504","0","-480"],["0","0","0","132","552","384","-384"],["0","0","0","0","144","288","-96"],["0","0","0","0","0","48","0"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","0","1","10","32","28","-32","-64"],["0","0","0","12","84","168","0","-160"],["0","0","0","0","44","184","128","-128"],["0","0","0","0","0","48","96","-32"],["0","0","0","0","0","0","16","0"]]},"deriv_order":2,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["2","16"]],"recurrence":{"denominator":{"coeffs":[["1","-14","40","44","-32","0"],["0","12","-92","8","128","-32"],["0","0","44","-104","-64","64"],["0","0","0","48","-32","-32"],["0","0","0","0","16","0"]]},"terms":[{"coeff":{"coeffs":[["1","2","-16","-36","0","64","0"],["0","12","44","-72","-256","-128","0"],["0","0","44","216","96","-288","-128"],["0","0","0","48","224","96","-128"],["0","0","0","0","16","64","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","-12","-32","-8","64","0","0"],["0","24","-48","-320","-288","128","0"],["0","0","88","112","-288","-352","0"],["0","0","0","96","192","-64","-128"],["0","0","0","0","32","64","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","0","0","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","8","272","4144","37248","218592","873216","2387776","4307968","4414976","493312","-5510144","-8101888","-5537792","-1900544","-262144"],["0","0","0","0","96","2976","41088","331968","1732096","6061440","14232576","21471488","17816064","1861632","-11569152","-11231232","-4423680","-655360"],["0","0","0","0","0","352","9920","122944","878464","3981952","11846912","23026432","27812352","17750016","1556480","-5242880","-3014656","-524288"],["0","0","0","0","0","0","384","9984","112640","720384","2860032","7244800","11550720","10905600","5167104","466944","-491520","-131072"],["0","0","0","0","0","0","0","128","3072","31488","179200","615936","1302528","1655808","1191936","442368","65536","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","24","816","12432","111744","655776","2619648","7163328","12923904","13244928","1479936","-16530432","-24305664","-16613376","-5701632","-786432"],["0","0","0","0","0","288","8928","123264","995904","5196288","18184320","42697728","64414464","53448192","5584896","-34707456","-33693696","-13271040","-1966080"],["0","0","0","0","0","0","1056","29760","368832","2635392","11945856","35540736","69079296","83437056","53250048","4669440","-15728640","-9043968","-1572864"],["0","0","0","0","0","0","0","1152","29952","337920","2161152","8580096","21734400","34652160","32716800","15501312","1400832","-1474560","-393216"],["0","0","0","0","0","0","0","0","384","9216","94464","537600","1847808","3907584","4967424","3575808","1327104","196608","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","8","272","4144","37248","218592","873216","2387776","4307968","4414976","493312","-5510144","-8101888","-5537792","-1900544","-262144"],["0","0","0","0","0","0","96","2976","41088","331968","1732096","6061440","14232576","21471488","17816064","1861632","-11569152","-11231232","-4423680","-655360"],["0","0","0","0","0","0","0","352","9920","122944","878464","3981952","11846912","23026432","27812352","17750016","1556480","-5242880","-3014656","-524288"],["0","0","0","0","0","0","0","0","384","9984","112640","720384","2860032","7244800","11550720","10905600","5167104","466944","-491520","-131072"],["0","0","0","0","0","0","0","0","0","128","3072","31488","179200","615936","1302528","1655808","1191936","442368","65536","0"]]},"deriv_order":2}],"shift":0}],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-10","-16"],["10","8"],["-4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-10","-46","-32"],["4","16","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-1","-8","-18","-8"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["2"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","224","6976","96704","782720","4070272","14045440","31778048","43452928","24333312","-22796288","-53026816","-40927232","-14811136","-2097152"],["0","0","0","0","0","0","1536","43136","532352","3783424","17025280","50106880","95619584","111178752","62942208","-6819840","-30752768","-15663104","-2621440"],["0","0","0","0","0","0","0","2048","53248","600576","3837952","15211520","38400000","60782592","56426496","25350144","901120","-3211264","-786432"],["0","0","0","0","0","0","0","0","768","18432","188928","1075200","3695616","7815168","9934848","7151616","2654208","393216","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","-112","-3808","-58016","-521472","-3060288","-12225024","-33428864","-60311552","-61809664","-6906368","77142016","113426432","77529088","26607616","3670016"],["0","0","0","0","0","0","-672","-20736","-285120","-2299392","-12038528","-42723840","-103921920","-170242048","-176424960","-94451712","2887680","34455552","17498112","2883584"],["0","0","0","0","0","0","0","-320","-9472","-124160","-948224","-4671488","-15536128","-35511296","-55836672","-59673600","-42250240","-18857984","-4784128","-524288"],["0","0","0","0","0","0","0","0","768","19968","224768","1428480","5594112","13772800","20637696","16601088","3710976","-3833856","-2752512","-524288"],["0","0","0","0","0","0","0","0","0","512","12288","125952","716800","2463744","5210112","6623232","4767744","1769472","262144","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","-48","-1632","-24864","-223488","-1311552","-5239296","-14326656","-25847808","-26489856","-2959872","33060864","48611328","33226752","11403264","1572864"],["0","0","0","0","0","0","0","-352","-10880","-149824","-1209088","-6322304","-22323200","-53617408","-85376000","-82563072","-33966080","16388096","26460160","11730944","1835008"],["0","0","0","0","0","0","0","0","-576","-16384","-205312","-1487360","-6866432","-20974592","-42539008","-55695360","-43557888","-16158720","704512","2424832","524288"],["0","0","0","0","0","0","0","0","0","-256","-6656","-75264","-484352","-1948672","-5068800","-8521728","-9007104","-5652480","-1900544","-262144","0"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","8","192","1968","11200","38496","81408","103488","74496","27648","4096"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","24","640","7440","49344","205088","552192","961728","1051392","678912","233472","32768"],["0","0","0","-16","-384","-3936","-22400","-76992","-162816","-206976","-148992","-55296","-8192","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","-24","-696","-8880","-65424","-307104","-956064","-1993536","-2752704","-2442240","-1320960","-393216","-49152"],["0","0","0","16","448","5504","38912","174464","515584","1012224","1302528","1065216","527360","143360","16384"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","-4","-128","-1824","-15232","-82528","-303360","-768640","-1337856","-1568832","-1197056","-563200","-147456","-16384"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","-8","-208","-2320","-14368","-53024","-113600","-112320","44160","237312","238592","102400","16384"],["0","0","-16","-384","-3936","-22400","-76992","-162816","-206976","-148992","-55296","-8192","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","4","104","1128","6352","17104","-3744","-187424","-655680","-1183872","-1245184","-757760","-245760","-32768"],["0","0","16","448","5504","38912","174464","515584","1012224","1302528","1065216","527360","143360","16384"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-4","-128","-1824","-15232","-82528","-303360","-768640","-1337856","-1568832","-1197056","-563200","-147456","-16384"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","0","8"],["2","-4","-8"],["0","4","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","2","28","24"],["0","-4","-16","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","8","18","8"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","128","3904","52864","416128","2089216","6874880","14495232","17457664","5846016","-14237696","-22257664","-14123008","-4325376","-524288"],["0","0","0","0","0","832","22784","272768","1868288","8035072","22355968","39731712","41994240","20090880","-4943872","-11091968","-5046272","-786432"],["0","0","0","0","0","0","896","22784","250624","1556992","5977600","14560256","22152192","19691520","8392704","114688","-1114112","-262144"],["0","0","0","0","0","0","0","256","6144","62976","358400","1231872","2605056","3311616","2383872","884736","131072","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","-64","-2176","-33152","-297984","-1748736","-6985728","-19102208","-34463744","-35319808","-3946496","44081152","64815104","44302336","15204352","2097152"],["0","0","0","0","0","-320","-9856","-135296","-1090304","-5716224","-20400640","-50304512","-84866048","-93861888","-60485632","-13500416","7995392","5767168","1048576"],["0","0","0","0","0","0","256","6912","81152","539136","2194944","5438464","7027712","-141312","-16115712","-26091520","-19562496","-7208960","-1048576"],["0","0","0","0","0","0","0","1024","26624","300032","1912832","7542784","18841600","29159424","25608192","9363456","-1933312","-2490368","-524288"],["0","0","0","0","0","0","0","0","512","12288","125952","716800","2463744","5210112","6623232","4767744","1769472","262144","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","-48","-1632","-24864","-223488","-1311552","-5239296","-14326656","-25847808","-26489856","-2959872","33060864","48611328","33226752","11403264","1572864"],["0","0","0","0","0","0","-352","-10880","-149824","-1209088","-6322304","-22323200","-53617408","-85376000","-82563072","-33966080","16388096","26460160","11730944","1835008"],["0","0","0","0","0","0","0","-576","-16384","-205312","-1487360","-6866432","-20974592","-42539008","-55695360","-43557888","-16158720","704512","2424832","524288"],["0","0","0","0","0","0","0","0","-256","-6656","-75264","-484352","-1948672","-5068800","-8521728","-9007104","-5652480","-1900544","-262144","0"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","8","192","1968","11200","38496","81408","103488","74496","27648","4096"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","8","208","2320","14368","53024","113600","112320","-44160","-237312","-238592","-102400","-16384"],["0","0","0","0","16","384","3936","22400","76992","162816","206976","148992","55296","8192","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","-4","-104","-1128","-6352","-17104","3744","187424","655680","1183872","1245184","757760","245760","32768"],["0","0","0","0","-16","-448","-5504","-38912","-174464","-515584","-1012224","-1302528","-1065216","-527360","-143360","-16384"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","4","128","1824","15232","82528","303360","768640","1337856","1568832","1197056","563200","147456","16384"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","16","440","5280","36176","155392","432288","776832","873408","576768","201728","28672"],["0","0","0","16","384","3936","22400","76992","162816","206976","148992","55296","8192"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","-8","-232","-2952","-21584","-99632","-299616","-581216","-682176","-384960","48128","194560","98304","16384"],["0","0","0","-16","-448","-5504","-38912","-174464","-515584","-1012224","-1302528","-1065216","-527360","-143360","-16384"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","4","128","1824","15232","82528","303360","768640","1337856","1568832","1197056","563200","147456","16384"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","0","8","192","1968","11200","38496","81408","103488","74496","27648","4096"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (2) theta + (1) theta^2; so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 2, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"f(n, theta) = 1 + (2) theta + (1) theta^2","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2, for every n >= 1"},"parent_proof":{"route":"liu_wang_three_term","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: coefficient of P_(n-1) must be 1 after normalization; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 3, with A_n(x) = (2) + (4) x, V_n(x) = 0, W_n(x) = (1) + (4) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 1 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 3: [x^0] W_n = 1 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 3: [x^0] W_n = 1 there (exact)","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 3, with A_n(x) = (2) + (4) x, B_n(x) = (1) + (4) x: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, -1/4] (so its coefficients have one sign); for n >= 2, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 3","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = A_n(x) P_(n-1) + B_n(x) P_(n-2) for n >= 3, with A_n(x) = (2) + (4) x, B_n(x) = (1) + (4) x","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, -1/4] (so its coefficients have one sign); for n >= 2, P_(n-1) interlaces P_n (weakly, in the sense of Liu and Wang), and deg P_n - deg P_(n-1) is 0 or 1 for n >= 3"],"theorem":"liu_wang_three_term: L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"liu_wang_three_term","rows_checked":10,"statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2 for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}