{"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"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"basis_map_A013609 of Pascal-(1,4,1) array."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-20","-50","-52","-84","-42","-4"],["-70","-140","63","127","14","0"],["20","100","142","18","0","0"],["25","45","0","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-70","-385","-907","-1101","-577","-88","-4"],["-145","-530","-622","-321","-64","-4","0"],["-5","0","37","8","0","0","0"],["25","45","0","0","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-50","-335","-855","-1017","-535","-84","-4"],["-75","-390","-685","-448","-78","-4","0"],["-25","-100","-105","-10","0","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["5","10","8","17","2"],["10","25","26","19","2"],["5","20","18","2","0"],["0","5","0","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["5","10","8","17","2"],["10","25","26","19","2"],["5","20","18","2","0"],["0","5","0","0","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0","0"],["0","0","-10","15","2"],["5","5","18","2","0"],["0","5","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["-45","-145","-27","193","56","4"],["35","205","221","-91","-14","0"],["55","35","-142","-18","0","0"],["-25","-45","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","45","275","579","457","80","4"],["25","190","475","428","78","4","0"],["25","100","105","10","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1","2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-2","-2"],["4","2"],["-5","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-15","-33","-6"],["10","20","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-5","-20","-21","-2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","125","1250","5250","12425","19535","23280","20868","11289","2502","236","8"],["0","0","375","3875","17000","41950","65955","71495","54114","25192","5244","480","16"],["0","0","375","4125","19250","49675","77605","75755","46044","16537","2982","252","8"],["0","0","125","1625","8500","23200","35485","30145","13218","2654","240","8","0"],["0","0","0","125","1000","3050","4300","2605","420","20","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","125","1250","5250","12425","19535","23280","20868","11289","2502","236","8"],["0","0","375","3875","17000","41950","65955","71495","54114","25192","5244","480","16"],["0","0","375","4125","19250","49675","77605","75755","46044","16537","2982","252","8"],["0","0","125","1625","8500","23200","35485","30145","13218","2654","240","8","0"],["0","0","0","125","1000","3050","4300","2605","420","20","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","125","1500","7575","20750","33015","30120","14361","2886","252","8"],["0","0","0","375","4500","22725","62250","99045","90360","43083","8658","756","24"],["0","0","0","375","4500","22725","62250","99045","90360","43083","8658","756","24"],["0","0","0","125","1500","7575","20750","33015","30120","14361","2886","252","8"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","125","2000","13575","51050","116015","162180","134841","60330","11796","1016","32"],["0","0","0","-250","-1500","2850","49400","182970","335940","332718","166560","34128","3008","96"],["0","0","0","-1500","-16500","-72900","-158100","-147180","34740","189108","137700","31608","2928","96"],["0","0","0","-1750","-20500","-100050","-260200","-379210","-289620","-80574","17040","8016","896","32"],["0","0","0","-625","-7500","-37875","-103750","-165075","-150600","-71805","-14430","-1260","-40","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-625","-9125","-57875","-208225","-465125","-662795","-595425","-321603","-96222","-14860","-1112","-32"],["0","0","0","0","-750","-9750","-54450","-169950","-322590","-378810","-266886","-103482","-18828","-1560","-48"],["0","0","0","1875","25875","154125","515775","1055475","1343205","1028655","431037","81702","6924","216","0"],["0","0","0","1250","17500","106000","362000","760300","1003000","812040","376320","88962","10892","664","16"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","-625","-10000","-70500","-287000","-742150","-1261800","-1409020","-1000200","-420801","-94408","-11224","-672","-16"],["0","0","0","-1250","-20000","-141000","-574000","-1484300","-2523600","-2818040","-2000400","-841602","-188816","-22448","-1344","-32"],["0","0","0","-625","-10000","-70500","-287000","-742150","-1261800","-1409020","-1000200","-420801","-94408","-11224","-672","-16"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","-625","-8750","-52625","-176500","-357425","-439250","-306975","-97800","-1398","3212","424","16"],["0","-1875","-26875","-165375","-567375","-1176025","-1482825","-1071525","-365205","-18624","8376","1232","48"],["0","-1875","-28125","-180375","-643125","-1383525","-1812975","-1372725","-508815","-47484","5856","1152","48"],["0","-625","-10625","-75125","-290125","-668675","-934475","-758775","-313215","-44688","-568","304","16"],["0","0","-625","-7500","-37875","-103750","-165075","-150600","-71805","-14430","-1260","-40","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","625","9375","61375","229125","533925","796675","746225","404775","99198","-1814","-3636","-440","-16"],["0","1250","20000","140250","564250","1429850","2353650","2495450","1621590","574716","85334","3620","-216","-16"],["0","625","11875","96375","441125","1257925","2317275","2752225","2028855","851838","176110","18148","888","16"],["0","0","1250","17500","106000","362000","760300","1003000","812040","376320","88962","10892","664","16"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","-625","-10000","-70500","-287000","-742150","-1261800","-1409020","-1000200","-420801","-94408","-11224","-672","-16"],["0","0","-1250","-20000","-141000","-574000","-1484300","-2523600","-2818040","-2000400","-841602","-188816","-22448","-1344","-32"],["0","0","-625","-10000","-70500","-287000","-742150","-1261800","-1409020","-1000200","-420801","-94408","-11224","-672","-16"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["5","5","-2"],["0","5","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-5","-5","12","4"],["0","-10","-20","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","5","20","21","2"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","-125","-1425","-6775","-16895","-22105","-10907","5899","8375","2250","228","8"],["0","-500","-4925","-19850","-40970","-41840","-9237","21562","19124","4732","464","16"],["0","-750","-6450","-21425","-32505","-15670","16432","25827","13143","2714","244","8"],["0","-500","-3825","-10400","-9680","5760","16947","10564","2414","232","8","0"],["0","-125","-875","-2050","-1250","1695","2185","400","20","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","125","1250","5250","12425","19535","23280","20868","11289","2502","236","8"],["0","375","3875","17000","41950","65955","71495","54114","25192","5244","480","16"],["0","375","4125","19250","49675","77605","75755","46044","16537","2982","252","8"],["0","125","1625","8500","23200","35485","30145","13218","2654","240","8","0"],["0","0","125","1000","3050","4300","2605","420","20","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","125","1500","7575","20750","33015","30120","14361","2886","252","8"],["0","0","0","375","4500","22725","62250","99045","90360","43083","8658","756","24"],["0","0","0","375","4500","22725","62250","99045","90360","43083","8658","756","24"],["0","0","0","125","1500","7575","20750","33015","30120","14361","2886","252","8"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","625","8750","52625","176500","357425","439250","306975","97800","1398","-3212","-424","-16"],["0","0","0","1875","26875","165375","567375","1176025","1482825","1071525","365205","18624","-8376","-1232","-48"],["0","0","0","1875","28125","180375","643125","1383525","1812975","1372725","508815","47484","-5856","-1152","-48"],["0","0","0","625","10625","75125","290125","668675","934475","758775","313215","44688","568","-304","-16"],["0","0","0","0","625","7500","37875","103750","165075","150600","71805","14430","1260","40","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-625","-9375","-61375","-229125","-533925","-796675","-746225","-404775","-99198","1814","3636","440","16"],["0","0","0","-1250","-20000","-140250","-564250","-1429850","-2353650","-2495450","-1621590","-574716","-85334","-3620","216","16"],["0","0","0","-625","-11875","-96375","-441125","-1257925","-2317275","-2752225","-2028855","-851838","-176110","-18148","-888","-16"],["0","0","0","0","-1250","-17500","-106000","-362000","-760300","-1003000","-812040","-376320","-88962","-10892","-664","-16"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","625","10000","70500","287000","742150","1261800","1409020","1000200","420801","94408","11224","672","16"],["0","0","0","0","1250","20000","141000","574000","1484300","2523600","2818040","2000400","841602","188816","22448","1344","32"],["0","0","0","0","625","10000","70500","287000","742150","1261800","1409020","1000200","420801","94408","11224","672","16"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","1250","17625","107250","366575","765900","994515","776130","330441","63126","5372","168"],["0","0","3750","53500","329250","1137600","2401450","3148620","2478990","1063128","203808","17376","544"],["0","0","3750","54750","344250","1213350","2608950","3478770","2780190","1206738","232668","19896","624"],["0","0","1250","19500","129750","480200","1077150","1489740","1227930","545856","106416","9152","288"],["0","0","0","625","7500","37875","103750","165075","150600","71805","14430","1260","40"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-1250","-19375","-131875","-516125","-1276075","-2058475","-2155245","-1404975","-519999","-92594","-7588","-232","0"],["0","0","-2500","-40000","-281250","-1138250","-2914150","-4877250","-5313490","-3621990","-1416318","-274150","-26068","-1128","-16"],["0","0","-1250","-21875","-166875","-728125","-2000075","-3579075","-4161245","-3029055","-1272639","-270518","-29372","-1560","-32"],["0","0","0","-1250","-17500","-106000","-362000","-760300","-1003000","-812040","-376320","-88962","-10892","-664","-16"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","625","10000","70500","287000","742150","1261800","1409020","1000200","420801","94408","11224","672","16"],["0","0","0","1250","20000","141000","574000","1484300","2523600","2818040","2000400","841602","188816","22448","1344","32"],["0","0","0","625","10000","70500","287000","742150","1261800","1409020","1000200","420801","94408","11224","672","16"]]},"deriv_order":2}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","25","200","610","860","521","84","4"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-4","-8"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-2"]]},"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 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 1 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"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"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":{"route":"liu_wang_three_term","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"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":"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 >= 2, with A_n(x) = (2) + (2) x, V_n(x) = 0, W_n(x) = (4) + (8) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 4 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 = 2: [x^0] W_n = 4 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 = 2: [x^0] W_n = 4 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 >= 2, with A_n(x) = (2) + (2) x, B_n(x) = (4) + (8) x: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/2] (so its coefficients have one sign); for n >= 1, 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 >= 2","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"interlacing":{"from_index":1,"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 >= 2, with A_n(x) = (2) + (2) x, B_n(x) = (4) + (8) x","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, -1/2] (so its coefficients have one sign); for n >= 1, 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 >= 2"],"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 >= 0, 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 = phi(n, D) = 1 + (1) D for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}