{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-24","0"],["15","-6"],["-2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-12","0"],["2","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2)."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["22","3","8","0"],["-22","51","9","4"],["4","-42","26","6"],["0","6","-21","0"],["0","0","2","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["8","38","7","16","0"],["-4","-2","115","29","12"],["0","-4","-34","82","22"],["0","0","2","-25","8"],["0","0","0","2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","4","8","2","4","0"],["0","-2","10","32","10","4"],["0","0","-4","4","28","8"],["0","0","0","-2","-2","4"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-2","-1","0"],["2","-1","1"],["0","2","-2"],["0","0","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-2","-1","0"],["0","2","-1","1"],["0","0","2","-2"],["0","0","0","1"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-4","2","-4"],["2","-5","8"],["0","2","-5"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["-48","96","-48","0"],["30","-153","114","0"],["-4","60","-101","6"],["0","-6","29","-8"],["0","0","-2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","6","24","-24","0"],["2","-18","-18","52","0"],["0","4","-10","-34","4"],["0","0","2","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":2,"multiplier":["-1","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-24","0"],["15","-6"],["-2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-12","0"],["2","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":2,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["-1/2","-9/4","-3","-1","0"],["1/2","7/4","5/4","0","1"],["0","1/2","3/2","0","-2"],["0","0","1/4","1","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-1/2","-9/4","-3","-1","0"],["0","1/2","7/4","5/4","0","1"],["0","0","1/2","3/2","0","-2"],["0","0","0","1/4","1","1"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["3/2"],["-5/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["7/4","7/2"],["-1/2","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["1/4","1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","4","16","16","0","0"],["0","-6","-24","-26","-8","-8"],["0","2","8","8","0","0"],["0","0","0","2","8","8"]]},"deriv_order":0}],"row_multiplier":["1/4","1","1"],"start_base":2,"step_multiplier":{"coeffs":[["1/16","1/2","3/2","2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["-7/16","-4","-29/2","-26","-23","-8"],["3/16","9/8","3/2","-3","-9","-6"],["-1/8","-7/8","-2","-1","2","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1/4","-21/8","-45/4","-25","-30","-18","-4"],["1/8","9/8","15/4","5","0","-6","-4"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["11","4"],["-11","2"],["2","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["4","10","4"],["-2","-2","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-1"],["1"]]},"deriv_order":0,"shift":1}]},"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 2) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (1) theta; so the candidate rows satisfy the input rule at every base n >= 2 (the exceptional bases checked exactly), and agreement on the 3 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, 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 + (1) theta","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-24","0"],["15","-6"],["-2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-12","0"],["2","2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta, for every n >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, 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 >= 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":"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":"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: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 3, with a_n(x) = ((-2 n^2 + 15 n - 24)/(n - 2)) + ((2 n^2 - 6 n)/(n - 2)) x and b_n(x) = ((2 n - 6)/(n - 2)) + ((2 n - 12)/(n - 2)) x + ((-4 n)/(n - 2)) x^2; open hypotheses: earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact); degree_step_positive: fails at n = 3: lambda_n = (-2 n^2 + 6 n)/(n - 2) = 0 there (exact); derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = 2 n - 6 is not the zero polynomial; derivative_coefficient_nonpositive: b_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] b_n <= 0 fails for k = 0 (not a disproof): fails at n = 4: [x^0] b_n = 1 there (exact); constant_term_positive: fails at n = 6: a_n(0) = (-2 n^2 + 15 n - 24)/(n - 2) = -3/2 there (exact)","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^2 + 15 n - 24)/(n - 2)) + ((2 n^2 - 6 n)/(n - 2)) x, V_n(x) = ((2 n - 6)/(n - 2)) + ((2 n - 12)/(n - 2)) x + ((-4 n)/(n - 2)) x^2, W_n(x) = 0; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); seed_degree_step: a seed row of the induction is zero; 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 = (-2 n^2 + 6 n)/(n - 2) = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_1(n, x) = 2 n - 6 is not the zero polynomial; constant_term_positive: fails at n = 6: A_n(0) = (-2 n^2 + 15 n - 24)/(n - 2) = -3/2 there (exact); derivative_coefficient_nonpositive: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 0 (not a disproof): fails at n = 4: [x^0] V_n = 1 there (exact); derivative_or_lag_coefficient_negative: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 0 (not a disproof): fails at n = 4: [x^0] V_n = 1 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): c_1 is not gamma(n) w(x)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_d(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a nonzero lag term and a nonzero P_(n-1) or P'_(n-1) term","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 3 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 3, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 6 n - 2 n^2 + 12 x - 3 n x + 18 x^2 - 12 n x^2 + 2 n^2 x^2, B = 4 n x + 12 x^2 - 2 n x^2 + 6 x^3 - 2 n x^3, C = 0) / (n - 2); open hypotheses: degree_step: delta = 2, deg P_(n-1) = 2 n - 6: fails at n = 3: Lambda(n, deg P_(n-1)) = -2 n^2 + 12 n - 18 = 0 there (exact); weyl_step_interlacing: at n = 3: A does not strictly interlace B (exact Bezout test); weyl_step_leading_coefficients_one_signed: fails at n = 4: A_a B_(a+1) = -4 n^3 + 36 n^2 - 108 n + 108 = -4 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, 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 + (1) theta for every n >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}