{"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":0,"initial_polynomials":[["1"],["2"],["0","5"],["0","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"veronese_section_2_0 of A natural number operator."},"identifying_rows":6,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","1376"],["0","0","0","135"],["0","0","0","-117"],["0","0","0","-18"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","9192"],["0","0","0","0","2493"],["0","0","0","0","-135"],["0","0","0","0","-54"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","7706"],["0","0","0","0","0","2745"],["0","0","0","0","0","153"],["0","0","0","0","0","-18"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","1266"],["0","0","0","0","0","0","522"],["0","0","0","0","0","0","54"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","468"],["0","0","558"],["0","0","270"],["0","0","36"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","0","0","-2964"],["0","0","0","-1350"],["0","0","0","378"],["0","0","0","108"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","0","0","0","-4992"],["0","0","0","0","-3222"],["0","0","0","0","-270"],["0","0","0","0","36"]]},"deriv_order":2,"shift":2},{"coeff":{"coeffs":[["0","0","0","0","0","-1092"],["0","0","0","0","0","-756"],["0","0","0","0","0","-108"]]},"deriv_order":3,"shift":2},{"coeff":{"coeffs":[["0","-420"],["0","-405"],["0","-153"],["0","-18"]]},"deriv_order":0,"shift":4},{"coeff":{"coeffs":[["0","0","-228"],["0","0","-279"],["0","0","-243"],["0","0","-54"]]},"deriv_order":1,"shift":4},{"coeff":{"coeffs":[["0","0","0","870"],["0","0","0","765"],["0","0","0","117"],["0","0","0","-18"]]},"deriv_order":2,"shift":4},{"coeff":{"coeffs":[["0","0","0","0","258"],["0","0","0","0","234"],["0","0","0","0","54"]]},"deriv_order":3,"shift":4},{"coeff":{"coeffs":[["16"]]},"deriv_order":0,"shift":6},{"coeff":{"coeffs":[["0","48"]]},"deriv_order":1,"shift":6},{"coeff":{"coeffs":[["0","0","16"]]},"deriv_order":2,"shift":6}]},"exceptional_bases":[0],"gauge":{"kind":"none"},"identifying_indices":[0,1,2,3,4,5],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["2"],["0","20"],["0","24"],["0","0","81"],["0","0","90"]],"recurrence":{"denominator":{"coeffs":[["16"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","-242"],["0","0","0","405"],["0","0","0","-207"],["0","0","0","18"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","-78"],["0","0","0","0","126"],["0","0","0","0","-54"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","0","936"],["0","0","-1206"],["0","0","378"],["0","0","-36"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","444"],["0","0","0","-540"],["0","0","0","108"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-390"],["0","513"],["0","-171"],["0","18"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-798"],["0","0","414"],["0","0","-54"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["0","0","0","0","1/16","1/2","3/2","2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2"],["0","5"],["0","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":4,"parent_valid_from_base":0,"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","-96878592"],["0","0","0","0","0","0","0","0","0","-909398016"],["0","0","0","0","0","0","0","0","0","-3450802176"],["0","0","0","0","0","0","0","0","0","-6737043456"],["0","0","0","0","0","0","0","0","0","-7115931648"],["0","0","0","0","0","0","0","0","0","-3917611008"],["0","0","0","0","0","0","0","0","0","-1003290624"],["0","0","0","0","0","0","0","0","0","-95551488"]]},"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","-22560768"],["0","0","0","0","0","0","0","0","0","0","-273051648"],["0","0","0","0","0","0","0","0","0","0","-1362604032"],["0","0","0","0","0","0","0","0","0","0","-3606405120"],["0","0","0","0","0","0","0","0","0","0","-5401313280"],["0","0","0","0","0","0","0","0","0","0","-4490919936"],["0","0","0","0","0","0","0","0","0","0","-1863254016"],["0","0","0","0","0","0","0","0","0","0","-286654464"]]},"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","299261952"],["0","0","0","0","0","0","0","0","0","0","0","2647904256"],["0","0","0","0","0","0","0","0","0","0","0","9251573760"],["0","0","0","0","0","0","0","0","0","0","0","15972360192"],["0","0","0","0","0","0","0","0","0","0","0","13793918976"],["0","0","0","0","0","0","0","0","0","0","0","5159780352"],["0","0","0","0","0","0","0","0","0","0","0","429981696"],["0","0","0","0","0","0","0","0","0","0","0","-95551488"]]},"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","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","85598208"],["0","0","0","0","0","0","0","0","0","0","0","0","762421248"],["0","0","0","0","0","0","0","0","0","0","0","0","2693357568"],["0","0","0","0","0","0","0","0","0","0","0","0","4745723904"],["0","0","0","0","0","0","0","0","0","0","0","0","4283891712"],["0","0","0","0","0","0","0","0","0","0","0","0","1815478272"],["0","0","0","0","0","0","0","0","0","0","0","0","286654464"]]},"deriv_order":3}],"shift":0},{"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","5308416"],["0","0","0","0","0","0","0","0","42467328"],["0","0","0","0","0","0","0","0","127401984"],["0","0","0","0","0","0","0","0","169869312"],["0","0","0","0","0","0","0","0","84934656"]]},"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","15925248"],["0","0","0","0","0","0","0","0","0","127401984"],["0","0","0","0","0","0","0","0","0","382205952"],["0","0","0","0","0","0","0","0","0","509607936"],["0","0","0","0","0","0","0","0","0","254803968"]]},"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","5308416"],["0","0","0","0","0","0","0","0","0","0","42467328"],["0","0","0","0","0","0","0","0","0","0","127401984"],["0","0","0","0","0","0","0","0","0","0","169869312"],["0","0","0","0","0","0","0","0","0","0","84934656"]]},"deriv_order":2}],"shift":2}],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","-4","0"],["0","-12","-20"],["0","16","-16"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","-8","40"],["0","0","-32","32"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-7"],["0","16"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"shift":3}]},"quotient":[],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","192"],["0","0","0","0","768"],["0","0","0","0","768"]]},"step_quotients":[[],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","960"],["0","0","0","0","0","4864"],["0","0","0","0","0","7936"],["0","0","0","0","0","4096"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","1920","-3456"],["0","0","0","0","0","11648","-23424"],["0","0","0","0","0","21248","-46848"],["0","0","0","0","0","11264","-27648"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","-3840","6912"],["0","0","0","0","0","0","-10752","16896"],["0","0","0","0","0","0","-6144","6144"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","5760","-10368"],["0","0","0","0","0","0","39168","-66816"],["0","0","0","0","0","17408","10240","-64512"],["0","0","0","0","0","51200","-155648","104448"],["0","0","0","0","0","32768","-131072","98304"]]},"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","-15360","43008","-27648"],["0","0","0","0","0","0","-43008","110592","-67584"],["0","0","0","0","0","0","-24576","49152","-24576"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["8","-15","0"],["0","4","-36"],["-8","32","-24"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","16","-56","72"],["0","16","-64","48"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-4","0"],["-8","24"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"shift":2}]},"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","-43794432"],["0","0","0","0","0","0","0","0","0","0","-439934976"],["0","0","0","0","0","0","0","0","0","0","-1825431552"],["0","0","0","0","0","0","0","0","0","0","-4025106432"],["0","0","0","0","0","0","0","0","0","0","-5048303616"],["0","0","0","0","0","0","0","0","0","0","-3567255552"],["0","0","0","0","0","0","0","0","0","0","-1305870336"],["0","0","0","0","0","0","0","0","0","0","-191102976"]]},"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","43794432"],["0","0","0","0","0","0","0","0","0","0","0","477757440"],["0","0","0","0","0","0","0","0","0","0","0","2110095360"],["0","0","0","0","0","0","0","0","0","0","0","4769611776"],["0","0","0","0","0","0","0","0","0","0","0","5669388288"],["0","0","0","0","0","0","0","0","0","0","0","3121348608"],["0","0","0","0","0","0","0","0","0","0","0","382205952"],["0","0","0","0","0","0","0","0","0","0","0","-127401984"]]},"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","-183140352"],["0","0","0","0","0","0","0","0","0","0","0","0","-1496973312"],["0","0","0","0","0","0","0","0","0","0","0","0","-4618321920"],["0","0","0","0","0","0","0","0","0","0","0","0","-6370099200"],["0","0","0","0","0","0","0","0","0","0","0","0","-3185049600"],["0","0","0","0","0","0","0","0","0","0","0","0","509607936"],["0","0","0","0","0","0","0","0","0","0","0","0","509607936"]]},"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","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-63700992"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-541458432"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-1783627776"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-2802843648"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-2038431744"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-509607936"]]},"deriv_order":3}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","576"],["0","0","0","0","2304"],["0","0","0","0","2304"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","576"],["0","0","0","0","2304"],["0","0","0","0","2304"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","-2304","4608"],["0","0","0","0","-10752","23040"],["0","0","0","0","-15360","36864"],["0","0","0","0","-6144","18432"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","-4608","17280","-17280"],["0","0","0","0","-26112","104832","-114048"],["0","0","0","0","-42240","186624","-221184"],["0","0","0","0","-16896","92160","-124416"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","9216","-34560","34560"],["0","0","0","0","0","23040","-87552","82944"],["0","0","0","0","0","9216","-36864","27648"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","-13824","51840","-51840"],["0","0","0","0","0","-89856","338688","-331776"],["0","0","0","0","-39936","62976","184320","-317952"],["0","0","0","0","-104448","546816","-940032","497664"],["0","0","0","0","-49152","344064","-737280","442368"]]},"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","36864","-175104","276480","-138240"],["0","0","0","0","0","92160","-442368","681984","-331776"],["0","0","0","0","0","36864","-184320","258048","-110592"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-4","-3"],["4","-12"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","6"],["0","-32","40"],["0","8","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","-16","16"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["2"],["4"]]},"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","-87588864","0"],["0","0","0","0","0","0","0","0","0","0","-879869952","0"],["0","0","0","0","0","0","0","0","0","0","-3650863104","-453869568"],["0","0","0","0","0","0","0","0","0","0","-8050212864","-2508226560"],["0","0","0","0","0","0","0","0","0","0","-10096607232","-3917611008"],["0","0","0","0","0","0","0","0","0","0","-7134511104","382205952"],["0","0","0","0","0","0","0","0","0","0","-2611740672","4586471424"],["0","0","0","0","0","0","0","0","0","0","-382205952","1911029760"]]},"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","87588864","0"],["0","0","0","0","0","0","0","0","0","0","0","955514880","3869835264"],["0","0","0","0","0","0","0","0","0","0","0","4220190720","22454599680"],["0","0","0","0","0","0","0","0","0","0","0","9539223552","40609382400"],["0","0","0","0","0","0","0","0","0","0","0","11338776576","14428274688"],["0","0","0","0","0","0","0","0","0","0","0","6242697216","-20448018432"],["0","0","0","0","0","0","0","0","0","0","0","764411904","-8790736896"],["0","0","0","0","0","0","0","0","0","0","0","-254803968","764411904"]]},"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","-366280704","0"],["0","0","0","0","0","0","0","0","0","0","0","0","-2993946624","4490919936"],["0","0","0","0","0","0","0","0","0","0","0","0","-9236643840","28474343424"],["0","0","0","0","0","0","0","0","0","0","0","0","-12740198400","62681776128"],["0","0","0","0","0","0","0","0","0","0","0","0","-6370099200","51980009472"],["0","0","0","0","0","0","0","0","0","0","0","0","1019215872","7644119040"],["0","0","0","0","0","0","0","0","0","0","0","0","1019215872","-3057647616"]]},"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","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-127401984","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-1082916864","764411904"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-3567255552","4968677376"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-5605687296","11466178560"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-4076863488","10701766656"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-1019215872","3057647616"]]},"deriv_order":3}],"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","1536"],["0","0","0","0","0","0","9216"],["0","0","0","0","0","0","18432"],["0","0","0","0","0","0","12288"]]},"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","3072"],["0","0","0","0","0","0","23552"],["0","0","0","0","0","0","69632"],["0","0","0","0","0","0","98304"],["0","0","0","0","0","0","65536"],["0","0","0","0","0","0","16384"]]},"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","6144","-11520"],["0","0","0","0","0","0","50176","-103680"],["0","0","0","0","0","0","154624","-347136"],["0","0","0","0","0","0","221184","-537600"],["0","0","0","0","0","0","143360","-380928"],["0","0","0","0","0","0","32768","-98304"]]},"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","-18432","29184"],["0","0","0","0","0","0","0","-86016","135168"],["0","0","0","0","0","0","0","-116736","184320"],["0","0","0","0","0","0","0","-24576","49152"],["0","0","0","0","0","0","0","24576","-24576"]]},"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","-12288","12288"],["0","0","0","0","0","0","0","0","-61440","61440"],["0","0","0","0","0","0","0","0","-98304","98304"],["0","0","0","0","0","0","0","0","-49152","49152"]]},"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","36864","-52992"],["0","0","0","0","0","0","12288","319488","-470016"],["0","0","0","0","0","0","126976","757760","-1281024"],["0","0","0","0","0","0","458752","32768","-933888"],["0","0","0","0","0","0","737280","-1818624","933888"],["0","0","0","0","0","0","524288","-1900544","1376256"],["0","0","0","0","0","0","131072","-524288","393216"]]},"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","18432","-18432"],["0","0","0","0","0","0","0","-98304","454656","-356352"],["0","0","0","0","0","0","0","-491520","1941504","-1449984"],["0","0","0","0","0","0","0","-786432","2899968","-2113536"],["0","0","0","0","0","0","0","-393216","1376256","-983040"]]},"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","-49152","98304","-49152"],["0","0","0","0","0","0","0","0","-245760","491520","-245760"],["0","0","0","0","0","0","0","0","-393216","786432","-393216"],["0","0","0","0","0","0","0","0","-196608","393216","-196608"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-1"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","6"],["0","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","4"]]},"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","0","0","0","0","-350355456","-650944512","0"],["0","0","0","0","0","0","0","0","0","0","-3519479808","-4518789120","-15479341056"],["0","0","0","0","0","0","0","0","0","0","-14603452416","-13791264768","-88002920448"],["0","0","0","0","0","0","0","0","0","0","-32200851456","-25066340352","-152404623360"],["0","0","0","0","0","0","0","0","0","0","-40386428928","-23410114560","-42042654720"],["0","0","0","0","0","0","0","0","0","0","-28538044416","4904976384","80263249920"],["0","0","0","0","0","0","0","0","0","0","-10446962688","25735200768","16817061888"],["0","0","0","0","0","0","0","0","0","0","-1528823808","10192158720","-10701766656"]]},"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","644972544","0"],["0","0","0","0","0","0","0","0","0","0","0","350355456","6533332992","0"],["0","0","0","0","0","0","0","0","0","0","0","3822059520","43778506752","-51406700544"],["0","0","0","0","0","0","0","0","0","0","0","16880762880","152563875840","-317613146112"],["0","0","0","0","0","0","0","0","0","0","0","38156894208","226011119616","-663891738624"],["0","0","0","0","0","0","0","0","0","0","0","45355106304","62745477120","-473553174528"],["0","0","0","0","0","0","0","0","0","0","0","24970788864","-114916589568","20639121408"],["0","0","0","0","0","0","0","0","0","0","0","3057647616","-46374322176","59624128512"],["0","0","0","0","0","0","0","0","0","0","0","-1019215872","4076863488","-3057647616"]]},"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","191102976","0"],["0","0","0","0","0","0","0","0","0","0","0","0","-1465122816","4236115968","0"],["0","0","0","0","0","0","0","0","0","0","0","0","-11975786496","45801013248","-27136622592"],["0","0","0","0","0","0","0","0","0","0","0","0","-36946575360","196326457344","-173521502208"],["0","0","0","0","0","0","0","0","0","0","0","0","-50960793600","369720557568","-388321247232"],["0","0","0","0","0","0","0","0","0","0","0","0","-25480396800","282322796544","-336341237760"],["0","0","0","0","0","0","0","0","0","0","0","0","4076863488","38730203136","-67268247552"],["0","0","0","0","0","0","0","0","0","0","0","0","4076863488","-16307453952","12230590464"]]},"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","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-509607936","509607936","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-4331667456","7389315072","-3057647616"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-14269022208","34143731712","-19874709504"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-22422749184","68287463424","-45864714240"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-16307453952","59114520576","-42807066624"],["0","0","0","0","0","0","0","0","0","0","0","0","0","-4076863488","16307453952","-12230590464"]]},"deriv_order":3}],"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","512"],["0","0","0","0","0","0","3072"],["0","0","0","0","0","0","6144"],["0","0","0","0","0","0","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","256"],["0","0","0","0","0","0","1280"],["0","0","0","0","0","0","2048"],["0","0","0","0","0","0","1024"]]},"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","512"],["0","0","0","0","0","0","0","1024"],["0","0","0","0","0","0","0","-2048"],["0","0","0","0","0","0","0","-4096"]]},"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","1024"],["0","0","0","0","0","0","0","0","4096"],["0","0","0","0","0","0","0","0","4096"]]},"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","-1536"],["0","0","0","0","0","0","-1024","-11520"],["0","0","0","0","0","0","-5120","-24576"],["0","0","0","0","0","0","-8192","-9216"],["0","0","0","0","0","0","-4096","12288"]]},"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","-1536"],["0","0","0","0","0","0","0","4096","-20480"],["0","0","0","0","0","0","0","14336","-61440"],["0","0","0","0","0","0","0","8192","-49152"],["0","0","0","0","0","0","0","-8192","8192"]]},"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","4096","-4096"],["0","0","0","0","0","0","0","0","16384","-16384"],["0","0","0","0","0","0","0","0","16384","-16384"]]},"deriv_order":2}]]}],"restart_bases":[0],"row_multiplier":{"coeffs":[["0","0","0","0","0","0","0","0","5308416"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":4}]},"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 0) 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 >= 0 (the exceptional bases checked exactly), and agreement on the 6 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"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 [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + 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":"0","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":0,"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":0,"initial_polynomials":[["1"],["2"],["0","5"],["0","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"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 >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, 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":"open"},"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: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","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":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a derivative or a lag term","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":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_d(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a nonzero lag term and a nonzero P_(n-1) or P'_(n-1) term","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2 for every n >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}