{"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"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","9"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"veronese_section_2_0 of Triangle of coefficients in expansion of (1+3*x)^n."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["-4","-136","936","-2916"],["0","42","1170","-4860"],["0","22","225","-2997"],["0","0","-54","-810"],["0","0","-9","-81"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-32","864","-7776","23328"],["0","-12","280","-7704","30132"],["0","0","302","-738","9396"],["0","0","66","-45","-2511"],["0","0","0","-234","-1782"],["0","0","0","-27","-243"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-40","1080","-9720","29160"],["0","0","-4","724","-12204","45684"],["0","0","0","262","-4140","25110"],["0","0","0","22","-675","5103"],["0","0","0","0","-144","0"],["0","0","0","0","-9","-81"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","-8","216","-1944","5832"],["0","0","0","0","172","-2628","9720"],["0","0","0","0","44","-1062","5994"],["0","0","0","0","0","-180","1620"],["0","0","0","0","0","-18","162"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","0"],["4","-8","36"],["0","-42","108"],["0","-22","117"],["0","0","54"],["0","0","9"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","0"],["0","12","-24","108"],["0","0","-126","324"],["0","0","-66","351"],["0","0","0","162"],["0","0","0","27"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","4","-8","36"],["0","0","0","-42","108"],["0","0","0","-22","117"],["0","0","0","0","54"],["0","0","0","0","9"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["-4","-12","0"],["4","10","0"],["0","24","-9"],["0","-22","-9"],["0","0","9"],["0","0","9"]]},"terms":[{"coeff":{"coeffs":[["-4","-156","36","-324"],["4","154","558","-648"],["0","24","-261","-81"],["0","-22","-351","567"],["0","0","9","405"],["0","0","9","81"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-88","216","-648"],["0","0","-84","972","-1944"],["0","0","-44","630","-2106"],["0","0","0","108","-972"],["0","0","0","18","-162"]]},"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":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","9"]]},"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"],["-1","-9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","18"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["1"]]},"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","64","-1280","8064","-20736","46656"],["0","0","0","-672","13824","-85536","139968"],["0","0","0","-352","8208","-62208","151632"],["0","0","0","0","864","-15552","69984"],["0","0","0","0","144","-2592","11664"]]},"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","192","-3840","24192","-62208","139968"],["0","0","0","0","-2016","41472","-256608","419904"],["0","0","0","0","-1056","24624","-186624","454896"],["0","0","0","0","0","2592","-46656","209952"],["0","0","0","0","0","432","-7776","34992"]]},"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","64","-1280","8064","-20736","46656"],["0","0","0","0","0","-672","13824","-85536","139968"],["0","0","0","0","0","-352","8208","-62208","151632"],["0","0","0","0","0","0","864","-15552","69984"],["0","0","0","0","0","0","144","-2592","11664"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0"],["0","0","0"],["0","-4","36"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0"],["0","-4","72","-324"],["0","-4","72","-324"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","8","-144","648"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","-4","72","-324"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-9"],["9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","54"],["0","-36"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-4","36"]]},"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","-1216","24768","-150336","233280"],["0","0","0","0","-800","19008","-147744","373248"],["0","0","0","0","0","2448","-44064","198288"],["0","0","0","0","0","432","-7776","34992"]]},"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","-512","11520","-82944","186624"],["0","0","0","0","0","-352","9792","-90720","279936"],["0","0","0","0","0","0","1728","-31104","139968"],["0","0","0","0","0","0","288","-5184","23328"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["0","0","16","-288","1296"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["0","0","144","-2592","11664"],["0","0","144","-2592","11664"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","-32","864","-7776","23328"],["0","0","0","-576","10368","-46656"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","-64","1728","-15552","46656"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","32","-864","7776","-23328"],["0","0","16","-144","-1296","11664"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","32","-864","7776","-23328"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","16","-288","1296"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","-9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-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":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","9"]]},"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":"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), V_n(x) = 0, W_n(x) = (-1) + (9) x; open hypotheses: seed_degree_step: degrees 0 and 0 (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 2: lambda_n = 0 = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -1 is not the zero polynomial","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), B_n(x) = (-1) + (9) x: 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) = (2), B_n(x) = (-1) + (9) x","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"},"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"}