{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","6"],["0","0","18"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","-1"],["0","0","0","0"],["0","0","0","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","6"],["0","0","3"],["0","0","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","6"],["0","-3"],["0","-6"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"theta_plus_one of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","-144","-264","-144","-24"],["0","0","0","-420","-574","-228","-26"],["0","0","0","-484","-483","-132","-9"],["0","0","0","-285","-197","-33","-1"],["0","0","0","-91","-39","-3","0"],["0","0","0","-15","-3","0","0"],["0","0","0","-1","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","-144","-264","-144","-24"],["0","0","0","-420","-574","-228","-26"],["0","0","0","-484","-483","-132","-9"],["0","0","0","-285","-197","-33","-1"],["0","0","0","-91","-39","-3","0"],["0","0","0","-15","-3","0","0"],["0","0","0","-1","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","216","180","36"],["0","0","216","702","393","57"],["0","0","522","822","291","24"],["0","0","465","429","87","3"],["0","0","195","102","9","0"],["0","0","39","9","0","0"],["0","0","3","0","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","216","180","36"],["0","0","216","702","393","57"],["0","0","522","822","291","24"],["0","0","465","429","87","3"],["0","0","195","102","9","0"],["0","0","39","9","0","0"],["0","0","3","0","0","0"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","-72","-96","-24"],["0","-72","-318","-240","-42"],["0","-222","-471","-210","-21"],["0","-255","-303","-75","-3"],["0","-135","-87","-9","0"],["0","-33","-9","0","0"],["0","-3","0","0","0"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","0","-72","-96","-24"],["0","-72","-318","-240","-42"],["0","-222","-471","-210","-21"],["0","-255","-303","-75","-3"],["0","-135","-87","-9","0"],["0","-33","-9","0","0"],["0","-3","0","0","0"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","12","18","6"],["12","58","51","11"],["40","96","51","6"],["51","71","21","1"],["31","24","3","0"],["9","3","0","0"],["1","0","0","0"]]},"deriv_order":0,"shift":3},{"coeff":{"coeffs":[["0","12","18","6"],["12","58","51","11"],["40","96","51","6"],["51","71","21","1"],["31","24","3","0"],["9","3","0","0"],["1","0","0","0"]]},"deriv_order":1,"shift":3}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["6","6"],["0","36","18"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0"],["-12","22","-12","2"],["40","-57","24","-3"],["-51","53","-15","1"],["31","-21","3","0"],["-9","3","0","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","-2","3","-1"],["0","0","0","-2","6","-3","0"],["0","0","0","3","-1","-3","1"],["0","0","0","1","-6","3","0"],["0","0","0","-3","3","0","0"],["0","0","0","1","0","0","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","18","-24","6"],["0","0","18","-39","6","3"],["0","0","-15","-15","21","-3"],["0","0","-15","33","-9","0"],["0","0","15","-9","0","0"],["0","0","-3","0","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","36","-30","6"],["0","36","-78","33","-3"],["0","-48","12","21","-6"],["0","-15","69","-33","3"],["0","45","-48","9","0"],["0","-21","9","0","0"],["0","3","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["24","164","490","839","906","639","294","85","14","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","6"],["0","0","18"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","-1"],["0","0","0","0"],["0","0","0","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","6"],["0","0","3"],["0","0","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","6"],["0","-3"],["0","-6"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":3,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","48","72","24"],["0","48","472","564","164"],["0","400","2036","1962","490"],["0","1496","5110","3987","839"],["0","3312","8316","5235","906"],["0","4819","9231","4635","639"],["0","4835","7140","2799","294"],["0","3411","3851","1137","85"],["0","1691","1420","297","14"],["0","577","341","45","1"],["0","129","48","3","0"],["0","17","3","0","0"],["0","1","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","48","72","24"],["0","48","472","564","164"],["0","400","2036","1962","490"],["0","1496","5110","3987","839"],["0","3312","8316","5235","906"],["0","4819","9231","4635","639"],["0","4835","7140","2799","294"],["0","3411","3851","1137","85"],["0","1691","1420","297","14"],["0","577","341","45","1"],["0","129","48","3","0"],["0","17","3","0","0"],["0","1","0","0","0"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0","-6"],["0","0","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-5"],["0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["2"],["1"]]},"deriv_order":0,"shift":2}]},"quotient":[{"coeff":{"coeffs":[["0","0","576","576"],["0","576","6096","4944"],["0","5520","29040","19152"],["0","24096","82536","44232"],["0","63384","156228","67764"],["0","111996","208005","72477"],["0","140241","200391","55437"],["0","127914","141399","30525"],["0","85962","73047","11997"],["0","42522","27273","3279"],["0","15276","7149","591"],["0","3870","1245","63"],["0","654","129","3"],["0","66","6","0"],["0","3","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-288","-288"],["0","0","-288","-2904","-2328"],["0","0","-2616","-13068","-8412"],["0","0","-10740","-34734","-17910"],["0","0","-26322","-60747","-24927"],["0","0","-42837","-73629","-23775"],["0","0","-48702","-63381","-15831"],["0","0","-39606","-39009","-7347"],["0","0","-23178","-17019","-2325"],["0","0","-9672","-5127","-477"],["0","0","-2802","-1011","-57"],["0","0","-534","-117","-3"],["0","0","-60","-6","0"],["0","0","-3","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["2"],["5"],["4"],["1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","48"],["0","208"],["0","364"],["0","328"],["0","160"],["0","40"],["0","4"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","-72"],["0","0","-258"],["0","0","-357"],["0","0","-237"],["0","0","-75"],["0","0","-9"]]},"deriv_order":0}],[]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","-6"],["0","-5"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"],["2"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","-2016","-288"],["0","0","-2016","-16296","-2904"],["0","0","-16008","-60612","-12492"],["0","0","-57708","-137610","-30654"],["0","0","-125118","-212721","-48003"],["0","0","-182067","-235653","-50553"],["0","0","-187650","-191151","-36603"],["0","0","-140598","-113745","-18237"],["0","0","-77142","-48945","-6129"],["0","0","-30708","-14751","-1323"],["0","0","-8622","-2937","-165"],["0","0","-1614","-345","-9"],["0","0","-180","-18","0"],["0","0","-9","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","2016","288"],["0","0","0","2016","14280","2616"],["0","0","0","13992","46332","9876"],["0","0","0","43716","91278","20778"],["0","0","0","81402","121443","27225"],["0","0","0","100665","114210","23328"],["0","0","0","86985","76941","13275"],["0","0","0","53613","36804","4962"],["0","0","0","23529","12141","1167"],["0","0","0","7179","2610","156"],["0","0","0","1443","327","9"],["0","0","0","171","18","0"],["0","0","0","9","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["2"],["5"],["4"],["1"]]},"step_quotients":[[{"coeff":{"coeffs":[["24"],["116"],["234"],["255"],["162"],["60"],["12"],["1"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","-24"],["0","-104"],["0","-182"],["0","-164"],["0","-80"],["0","-20"],["0","-2"]]},"deriv_order":0}],[]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","-3456","5184","-1728"],["0","0","0","-22176","33264","-11088"],["0","0","0","-62448","93672","-31224"],["0","0","0","-101496","152244","-50748"],["0","0","0","-105324","157986","-52662"],["0","0","0","-72864","109296","-36432"],["0","0","0","-34020","51030","-17010"],["0","0","0","-10584","15876","-5292"],["0","0","0","-2100","3150","-1050"],["0","0","0","-240","360","-120"],["0","0","0","-12","18","-6"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","1"],["0","4"],["0","6"],["0","4"],["0","1"]]},"step_quotients":[[],[{"coeff":{"coeffs":[["0","-6"],["0","-29"],["0","-57"],["0","-58"],["0","-32"],["0","-9"],["0","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","6"],["0","0","23"],["0","0","34"],["0","0","24"],["0","0","8"],["0","0","1"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","36"],["0","0","168"],["0","0","325"],["0","0","337"],["0","0","202"],["0","0","70"],["0","0","13"],["0","0","1"]]},"deriv_order":0}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","-24"],["0","0","0","-26"],["0","0","0","-9"],["0","0","0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","36"],["0","0","57"],["0","0","24"],["0","0","3"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-24"],["0","-42"],["0","-21"],["0","-3"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["6"],["11"],["6"],["1"]]},"deriv_order":0,"shift":3}]},"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 = phi(n, D) = 1 + (1) D; so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 3 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.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted 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":"phi(n, D) = 1 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","6"],["0","0","18"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","-1"],["0","0","0","0"],["0","0","0","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","6"],["0","0","3"],["0","0","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","6"],["0","-3"],["0","-6"],["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, 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 differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"verdict":"open"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","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: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","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 = phi(n, D) = 1 + (1) D 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"}
