{"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"],["1","3","2"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","3","2"],["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Scaled array A078740 ((3,2)-Stirling2)."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["6","72","90","-228","-468","-216"],["11","54","-123","-634","-756","-288"],["6","-15","-277","-592","-432","-96"],["1","-29","-140","-206","-96","0"],["0","-9","-29","-20","0","0"],["0","-1","-1","0","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","54","240","-162","-1764","-2352","-936"],["0","61","42","-1257","-3590","-3600","-1248"],["0","26","-205","-1495","-2720","-1872","-416"],["0","3","-143","-612","-834","-368","0"],["0","0","-35","-103","-68","0","0"],["0","0","-3","-3","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","51","102","-450","-1578","-1653","-576"],["0","0","46","-96","-1233","-2744","-2421","-768"],["0","0","16","-215","-1107","-1772","-1152","-256"],["0","0","1","-99","-380","-476","-196","0"],["0","0","0","-19","-49","-30","0","0"],["0","0","0","-1","-1","0","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","9","6","-108","-270","-237","-72"],["0","0","0","7","-30","-222","-422","-333","-96"],["0","0","0","2","-40","-166","-236","-144","-32"],["0","0","0","0","-14","-48","-54","-20","0"],["0","0","0","0","-2","-4","-2","0","0"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["-6","0","18","12"],["-11","12","81","52"],["-6","43","100","64"],["-1","31","55","16"],["0","9","10","0"],["0","1","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-18","0","54","36"],["0","-33","36","243","156"],["0","-18","129","300","192"],["0","-3","93","165","48"],["0","0","27","30","0"],["0","0","3","0","0"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","-6","0","18","12"],["0","0","-11","12","81","52"],["0","0","-6","43","100","64"],["0","0","-1","31","55","16"],["0","0","0","9","10","0"],["0","0","0","1","0","0"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","8","-8","8"],["-2","-12","6","-28"],["-3","-6","-5","16"],["-1","5","15","16"],["0","4","10","0"],["0","1","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-24","-48","0","48","24"],["-2","-28","-122","-12","180","96"],["-3","-28","21","94","144","96"],["-1","3","34","126","96","0"],["0","4","24","20","0","0"],["0","1","1","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-8","8","32","28","8"],["0","-3","-16","18","104","105","32"],["0","-2","10","46","86","84","32"],["0","0","6","32","46","20","0"],["0","0","2","4","2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["27","27","9","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"],["1","3","2"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","3","2"],["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"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":[["-3","-9","-6"],["-1","-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","-4","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","-54","-216","-162","540","1350","1296","594","108"],["0","0","-135","-432","459","3996","7479","6696","2997","540"],["0","0","-126","-45","2484","8766","13734","11475","5004","900"],["0","0","-56","325","3036","8354","11516","8733","3496","580"],["0","0","-12","262","1688","4052","4988","3350","1160","160"],["0","0","-1","90","485","1036","1129","650","179","16"],["0","0","0","15","70","130","120","55","10","0"],["0","0","0","1","4","6","4","1","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-162","-648","-486","1620","4050","3888","1782","324"],["0","0","0","-405","-1296","1377","11988","22437","20088","8991","1620"],["0","0","0","-378","-135","7452","26298","41202","34425","15012","2700"],["0","0","0","-168","975","9108","25062","34548","26199","10488","1740"],["0","0","0","-36","786","5064","12156","14964","10050","3480","480"],["0","0","0","-3","270","1455","3108","3387","1950","537","48"],["0","0","0","0","45","210","390","360","165","30","0"],["0","0","0","0","3","12","18","12","3","0","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","-54","-216","-162","540","1350","1296","594","108"],["0","0","0","0","-135","-432","459","3996","7479","6696","2997","540"],["0","0","0","0","-126","-45","2484","8766","13734","11475","5004","900"],["0","0","0","0","-56","325","3036","8354","11516","8733","3496","580"],["0","0","0","0","-12","262","1688","4052","4988","3350","1160","160"],["0","0","0","0","-1","90","485","1036","1129","650","179","16"],["0","0","0","0","0","15","70","130","120","55","10","0"],["0","0","0","0","0","1","4","6","4","1","0","0"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","9","18","9"],["0","6","12","6"],["0","1","2","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","12","84","180","156","48"],["0","19","133","285","247","76"],["0","8","56","120","104","32"],["0","1","7","15","13","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-24","-96","-144","-96","-24"],["0","0","-14","-56","-84","-56","-14"],["0","0","-2","-8","-12","-8","-2"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","-48","-192","-288","-192","-48"],["0","0","-28","-112","-168","-112","-28"],["0","0","-4","-16","-24","-16","-4"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-5","-6"],["-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["2","6","4"],["0","-2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","1","2","1"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","270","918","-1458","-10476","-18414","-15066","-5886","-864"],["0","0","0","324","-297","-9828","-32454","-48816","-38745","-15768","-2592"],["0","0","0","144","-1611","-12852","-34434","-46476","-34155","-13032","-2016"],["0","0","0","28","-1154","-7164","-17164","-21076","-13986","-4700","-608"],["0","0","0","2","-360","-1966","-4288","-4762","-2776","-762","-64"],["0","0","0","0","-53","-256","-494","-476","-229","-44","0"],["0","0","0","0","-3","-12","-18","-12","-3","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","135","216","-2673","-11880","-21195","-19440","-9099","-1728"],["0","0","0","0","162","-648","-8856","-29160","-46710","-40608","-18468","-3456"],["0","0","0","0","72","-1278","-9828","-27396","-39384","-31374","-13212","-2304"],["0","0","0","0","14","-820","-5130","-12680","-16310","-11556","-4270","-640"],["0","0","0","0","1","-248","-1363","-3016","-3429","-2080","-617","-64"],["0","0","0","0","0","-36","-174","-336","-324","-156","-30","0"],["0","0","0","0","0","-2","-8","-12","-8","-2","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","27","108","162","108","27"],["0","0","27","108","162","108","27"],["0","0","9","36","54","36","9"],["0","0","1","4","6","4","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","-216","-1080","-2160","-2160","-1080","-216"],["0","0","-450","-2214","-4356","-4284","-2106","-414"],["0","0","-327","-1566","-2994","-2856","-1359","-258"],["0","0","-109","-502","-918","-832","-373","-66"],["0","0","-17","-74","-126","-104","-41","-6"],["0","0","-1","-4","-6","-4","-1","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","72","432","1080","1440","1080","432","72"],["0","0","66","324","630","600","270","36","-6"],["0","0","20","54","-30","-260","-360","-210","-46"],["0","0","2","-8","-70","-160","-170","-88","-18"],["0","0","0","-2","-10","-20","-20","-10","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","36","216","540","720","540","216","36"],["0","0","0","33","198","495","660","495","198","33"],["0","0","0","10","60","150","200","150","60","10"],["0","0","0","1","6","15","20","15","6","1"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","180","1188","3240","4680","3780","1620","288"],["0","0","201","1269","3330","4650","3645","1521","264"],["0","0","83","495","1230","1630","1215","483","80"],["0","0","15","83","190","230","155","55","8"],["0","0","1","5","10","10","5","1","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","72","432","1080","1440","1080","432","72"],["0","0","0","66","396","990","1320","990","396","66"],["0","0","0","20","120","300","400","300","120","20"],["0","0","0","2","12","30","40","30","12","2"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","1","4","6","4","1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","2","1"],["1","2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-5","-15","-10"],["-2","-6","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["4"],["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 = 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 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.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"],["1","3","2"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","3","2"],["2","6","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"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":{"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 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 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: c_1(n, x) has degree > 1 in x","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: deg_x c_0 > 1, deg_x c_1 > 2 or deg_x c_2 > 2: the degree may grow by more than one","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":"Q_n = A_n(x) Q_(n-1) + B_n(x) Q_(n-2) for n >= 2, with A_n(x) = ((2 n + 1)/(n + 2)) + ((4 n + 2)/(n + 2)) x, B_n(x) = ((-n + 1)/(n + 2)); Q_n = P_n / (x + 1)^(n - 0) for n >= 0, which satisfies the rule below: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); for n >= 1, Q_(n-1) interlaces Q_n (weakly, in the sense of Liu and Wang; no interlacing of the P_n is claimed)","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"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 = f(n, theta) = 1 + (2) theta + (1) theta^2 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"}