{"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":[["2"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["-800","-3530","-5098","-3268","-1048","-176"],["0","-2700","-8880","-9892","-4500","-720"],["0","0","-3000","-6584","-4608","-1016"],["0","0","0","-1120","-1584","-544"],["0","0","0","0","-128","-96"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["500","3300","8740","11996","9048","3568","576"],["0","600","5210","15746","19172","9832","1712"],["0","0","-3300","-9040","-5948","436","848"],["0","0","0","-6600","-13512","-8448","-1512"],["0","0","0","0","-3040","-4176","-1376"],["0","0","0","0","0","-384","-288"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","625","4125","10925","14995","11310","4460","720"],["0","0","2950","16220","33702","32952","15172","2624"],["0","0","0","3300","13120","19768","12920","3040"],["0","0","0","0","0","1216","2112","904"],["0","0","0","0","0","-720","-864","-224"],["0","0","0","0","0","0","-128","-96"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","125","825","2185","2999","2262","892","144"],["0","0","0","750","3950","7760","7244","3244","560"],["0","0","0","0","1200","4400","5932","3484","752"],["0","0","0","0","0","600","1560","1344","384"],["0","0","0","0","0","0","80","144","64"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["200","370","202","36","8"],["0","500","720","328","48"],["0","0","600","536","104"],["0","0","0","240","96"],["0","0","0","0","32"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","600","1110","606","108","24"],["0","0","1500","2160","984","144"],["0","0","0","1800","1608","312"],["0","0","0","0","720","288"],["0","0","0","0","0","96"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","200","370","202","36","8"],["0","0","0","500","720","328","48"],["0","0","0","0","600","536","104"],["0","0","0","0","0","240","96"],["0","0","0","0","0","0","32"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["2"],["3","8"]],"recurrence":{"denominator":{"coeffs":[["-200","130","-82","-4","0"],["200","-630","562","28","0"],["0","500","-1080","160","-8"],["0","0","600","-424","40"],["0","0","0","240","-64"],["0","0","0","0","32"]]},"terms":[{"coeff":{"coeffs":[["-800","-830","782","1160","300","-24"],["800","-1870","-3662","-1244","176","88"],["0","2700","-120","-3140","-1100","-24"],["0","0","3000","2104","-448","-200"],["0","0","0","1120","944","64"],["0","0","0","0","128","96"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-125","-75","565","961","530","68","-16"],["0","-750","-1550","-760","260","268","48"],["0","0","-1200","-2600","-1732","-316","16"],["0","0","0","-600","-1240","-768","-128"],["0","0","0","0","-80","-144","-64"]]},"deriv_order":1,"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":0,"initial_polynomials":[["2"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-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":[["0","0"],["-8","-6"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["5","9","4"]]},"deriv_order":1,"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"],["20000","109000","250200","313000","230888","102160","27168","4608","512"],["0","50000","252000","534000","615360","415392","163776","34816","3072"],["0","0","60000","269600","493760","469664","243104","64256","6656"],["0","0","0","24000","96000","150720","115584","43008","6144"],["0","0","0","0","3200","11520","15488","9216","2048"]]},"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","60000","327000","750600","939000","692664","306480","81504","13824","1536"],["0","0","150000","756000","1602000","1846080","1246176","491328","104448","9216"],["0","0","0","180000","808800","1481280","1408992","729312","192768","19968"],["0","0","0","0","72000","288000","452160","346752","129024","18432"],["0","0","0","0","0","9600","34560","46464","27648","6144"]]},"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","20000","109000","250200","313000","230888","102160","27168","4608","512"],["0","0","0","50000","252000","534000","615360","415392","163776","34816","3072"],["0","0","0","0","60000","269600","493760","469664","243104","64256","6656"],["0","0","0","0","0","24000","96000","150720","115584","43008","6144"],["0","0","0","0","0","0","3200","11520","15488","9216","2048"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0"],["0","0","0"],["20","36","16"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0"],["-20","-56","-52","-16"],["-20","-56","-52","-16"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["50","180","242","144","32"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0"],["-50","-180","-242","-144","-32"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-2"],["4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["7","6"],["-10","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["5","9","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","-35000","-141000","-153400","137000","491440","503456","251712","60416","5120"],["0","0","-30000","26000","491200","1146960","1221520","676416","186112","19456"],["0","0","0","-8000","55200","301280","518880","418784","162048","24064"],["0","0","0","0","0","14400","51840","69696","41472","9216"]]},"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","-15000","-67000","-99400","-9320","145440","192352","115200","34304","4096"],["0","0","0","-20000","-40000","84800","359520","484256","324000","109312","14848"],["0","0","0","0","-12000","-7200","98720","237600","227648","101376","17408"],["0","0","0","0","0","0","9600","34560","46464","27648","6144"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["200","720","968","576","128"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["200","720","968","576","128"],["600","2160","2904","1728","384"],["400","1440","1936","1152","256"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["-300","-1280","-2172","-1832","-768","-128"],["-1000","-4400","-7720","-6752","-2944","-512"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["500","2700","6060","7236","4848","1728","256"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["100","360","484","288","64","0"],["800","3480","6032","5208","2240","384"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["-500","-2700","-6060","-7236","-4848","-1728","-256"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["200","720","968","576","128"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","-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 = 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 >= 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":[["2"],["3","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-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 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":"Unresolved: P_n = (a_n + b_n (x + 1)) P_(n-1) - c_n (x + 1)^2 P_(n-2) for n >= 2, with a_n = 1, b_n = 2, c_n = 1; Q_n(y) = y^(d_n) P_n(1/y - 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: root_region: the region contains the pole x = -1 of the coordinate change y = 1/(x + 1)","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) = (3) + (2) x, V_n(x) = 0, W_n(x) = (-1) + (-2) x + (-1) x^2; 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 = 1 (not a disproof): fails at n = 2: [x^1] W_n = -2 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 = 1 (not a disproof): fails at n = 2: [x^1] W_n = -2 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) = (3) + (2) x, B_n(x) = (-1) + (-2) x + (-1) x^2: 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, 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) = (3) + (2) x, B_n(x) = (-1) + (-2) x + (-1) x^2","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, 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; Proposition 3.3 (the case B_n(x) = -(c_n x + d_n)^2)"},"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 = 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"}