{"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":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["3"],["-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","8"],["1","-4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","16","-8","16"],["-1","-18","-44","32"],["0","0","24","4"],["0","4","27","-28"],["0","0","-2","-20"],["0","0","-1","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-8","96","-120","176"],["0","-3","10","-484","480"],["0","0","32","-160","428"],["0","0","12","33","108"],["0","0","0","-14","-28"],["0","0","0","-3","-12"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-10","76","-128","176"],["0","0","-1","62","-484","512"],["0","0","0","40","-266","524"],["0","0","0","4","-33","212"],["0","0","0","0","-12","20"],["0","0","0","0","-1","-4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","-2","12","-24","32"],["0","0","0","0","16","-88","96"],["0","0","0","0","8","-58","104"],["0","0","0","0","0","-12","48"],["0","0","0","0","0","-2","8"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","0"],["1","2","0"],["0","0","0"],["0","-4","1"],["0","0","2"],["0","0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","0"],["0","3","6","0"],["0","0","0","0"],["0","0","-12","3"],["0","0","0","6"],["0","0","0","3"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","1","2","0"],["0","0","0","0","0"],["0","0","0","-4","1"],["0","0","0","0","2"],["0","0","0","0","1"]]},"deriv_order":2,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1","2","0"],["1","-10","0"],["0","12","-1"],["0","-4","3"],["0","0","-3"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-1","-30","-32","0"],["1","6","8","0"],["0","12","59","-8"],["0","-4","-25","-12"],["0","0","-3","0"],["0","0","1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-4","-16","0"],["0","0","0","0","0"],["0","0","-8","34","-8"],["0","0","0","4","-16"],["0","0","0","2","-8"]]},"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":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["3"],["-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","8"],["1","-4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":1,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","4"],["-1","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","8"]]},"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","1","-6","0","32","0"],["0","0","0","0","0","0","0"],["0","0","0","-4","33","-72","16"],["0","0","0","0","2","-16","32"],["0","0","0","0","1","-8","16"]]},"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","3","-18","0","96","0"],["0","0","0","0","0","0","0","0"],["0","0","0","0","-12","99","-216","48"],["0","0","0","0","0","6","-48","96"],["0","0","0","0","0","3","-24","48"]]},"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","1","-6","0","32","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","-4","33","-72","16"],["0","0","0","0","0","0","2","-16","32"],["0","0","0","0","0","0","1","-8","16"]]},"deriv_order":2}],"step_multiplier":{"coeffs":[["0","0","0"],["0","0","0"],["0","-1","4"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","0"],["0","1","-8","16"],["0","1","-8","16"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","-2","16","-32"],["0","0","-2","16","-32"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","4","-32","64"],["0","0","4","-32","64"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["2"],["-3"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","10"],["0","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-1","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","-20","168","-384","128"],["0","0","0","0","-36","316","-800","448"],["0","0","0","0","0","32","-256","512"],["0","0","0","0","0","12","-96","192"]]},"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","-8","72","-192","128"],["0","0","0","0","0","-16","152","-448","384"],["0","0","0","0","0","0","24","-192","384"],["0","0","0","0","0","0","8","-64","128"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["0","0","1","-8","16"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","0"],["0","0","1","-8","16"],["0","0","0","0","0"],["0","0","-1","8","-16"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","1","-14","64","-96"],["0","0","1","-10","32","-32"],["0","0","0","4","-32","64"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","1","-12","48","-64"],["0","0","0","1","-12","48","-64"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","-2","28","-128","192"],["0","0","-3","32","-112","128"],["0","0","-1","4","16","-64"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","-2","24","-96","128"],["0","0","0","-2","24","-96","128"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","1","-8","16"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["-1","4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"],["2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-1"],["-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 1) 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 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"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 [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + 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":"1","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":1,"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":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["3"],["-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","8"],["1","-4"]]},"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 >= 1"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 1, 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 >= 1, 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":"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":"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_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 >= 3, with A_n(x) = ((2 n - 3)/(n - 1)), V_n(x) = 0, W_n(x) = ((-n + 2)/(n - 1)) + ((4 n - 8)/(n - 1)) 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 = 3: lambda_n = (0)/(n - 1) = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = n - 2 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":"the parent equals the flagged entry, but the flag gives real-rootedness only and one-signed coefficients are not shown: the leader is not positive for every n >= 3","route":"attestation","success":false,"variant":"oeis_catalog:A089627:lean_proved"},{"detail":"no lean_proved catalog entry equals the parent; 2 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A052553:lean_proved, oeis_catalog:A334218:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 1, 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 = f(n, theta) = 1 + (2) theta + (1) theta^2 for every n >= 1 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}